{"id":139,"date":"2021-02-03T21:52:41","date_gmt":"2021-02-03T21:52:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/?post_type=chapter&p=139"},"modified":"2021-04-19T17:13:42","modified_gmt":"2021-04-19T17:13:42","slug":"problem-set-trigonometric-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/problem-set-trigonometric-functions\/","title":{"raw":"Problem Set: Trigonometric Functions","rendered":"Problem Set: Trigonometric Functions"},"content":{"raw":"

For the following exercises (1-5), convert each angle in degrees to radians. Write the answer as a multiple of [latex]\\pi[\/latex].<\/p>\r\n\r\n

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1.\u00a0<\/strong>240\u00b0<\/p>\r\n[reveal-answer q=\"fs-id1170572433916\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572433916\"]\r\n

[latex]\\frac{4\\pi}{3}[\/latex] rad<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.\u00a0<\/strong>15\u00b0<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>-60\u00b0<\/p>\r\n[reveal-answer q=\"fs-id1170572222350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572222350\"]\r\n

[latex]\\frac{\u2212\\pi }{3}[\/latex] rad<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4.\u00a0<\/strong>-225\u00b0<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>330\u00b0<\/p>\r\n[reveal-answer q=\"fs-id1170572227995\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572227995\"]\r\n

[latex]\\frac{11\\pi}{6}[\/latex] rad<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (6-10), convert each angle in radians to degrees.<\/p>\r\n\r\n

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6.\u00a0<\/strong>[latex]\\large \\frac{\\pi}{2}[\/latex] rad<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7.\u00a0<\/strong>[latex]\\large \\frac{7\\pi}{6}[\/latex] rad<\/p>\r\n[reveal-answer q=\"fs-id1170572242130\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572242130\"]\r\n

210\u00b0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>[latex]\\large\\frac{11\\pi}{2}[\/latex] rad<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]-3\\pi[\/latex] rad<\/p>\r\n[reveal-answer q=\"fs-id1170572480661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480661\"]\r\n

-540\u00b0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10.\u00a0<\/strong>[latex]\\large\\frac{5\\pi}{12}[\/latex] rad<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

Evaluate the following functional values (11-16).<\/p>\r\n\r\n

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11.\u00a0<\/strong>[latex]\\cos \\Big(\\large\\frac{4\\pi}{3}\\Big)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572363409\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572363409\"]\r\n

-1\/2<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12.\u00a0<\/strong>[latex]\\tan \\Big(\\large\\frac{19\\pi}{4}\\Big)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>[latex]\\sin\\left(-\\large\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572228334\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572228334\"]\r\n

[latex]-\\large\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]\\sec\\Big(\\large\\frac{\\pi}{6}\\Big)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>[latex]\\sin\\Big(\\large\\frac{\\pi}{12}\\Big)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572547496\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572547496\"]\r\n

[latex]\\large \\frac{\\sqrt{3}-1}{2\\sqrt{2}} \\normalsize = \\large \\frac{\\sqrt{6}-\\sqrt{2}}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]\\cos \\Big(\\large \\frac{5\\pi}{12}\\Big)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (17-22), consider triangle [latex]ABC[\/latex], a right triangle with a right angle at [latex]C[\/latex].\u00a0(a)<\/strong> Find the missing side of the triangle, and\u00a0(b)<\/strong>\u00a0find the six trigonometric function values for the angle at [latex]A[\/latex]. Where necessary, round to one decimal place.<\/p>\r\n\"An<\/span>\r\n

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17.\u00a0<\/strong>[latex]a=4, \\, c=7[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572552366\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572552366\"]\r\n

a. [latex]b=5.7[\/latex] b. [latex]\\sin A=\\frac{4}{7}, \\, \\cos A=\\frac{5.7}{7}, \\, \\tan A=\\frac{4}{5.7}, \\, \\csc A=\\frac{7}{4}, \\, \\sec A=\\frac{7}{5.7}, \\, \\cot A=\\frac{5.7}{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]a=21, \\, c=29[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0<\/strong>[latex]a=85.3, \\, b=125.5[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572244361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572244361\"]\r\n

a. [latex]c=151.7[\/latex] b. [latex]\\sin A=0.5623, \\, \\cos A=0.8273, \\, \\tan A=0.6797, \\, \\csc A=1.778, \\, \\sec A=1.209, \\, \\cot A=1.471[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]b=40, \\, c=41[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]a=84, \\, b=13[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572142044\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572142044\"]\r\n

a. [latex]c=85[\/latex] b. [latex]\\sin A=\\frac{84}{85}, \\, \\cos A=\\frac{13}{85}, \\, \\tan A=\\frac{84}{13}, \\, \\csc A=\\frac{85}{84}, \\, \\sec A=\\frac{85}{13}, \\, \\cot A=\\frac{13}{84}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]b=28, \\, c=35[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (23-26), [latex]P[\/latex] is a point on the unit circle. (a)<\/strong> Find the (exact) missing coordinate value of each point and (b)<\/strong> find the values of the six trigonometric functions for the angle [latex]\\theta [\/latex] with a terminal side that passes through point [latex]P[\/latex]. Rationalize all denominators.<\/p>\r\n\r\n

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23.\u00a0<\/strong>[latex]P\\left(\\frac{7}{25},y\\right), \\, y>0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572546952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572546952\"]\r\n

a. [latex]y=\\frac{24}{25}[\/latex] b. [latex]\\sin \\theta =\\frac{24}{25}, \\, \\cos \\theta =\\frac{7}{25}, \\, \\tan \\theta =\\frac{24}{7}, \\, \\csc \\theta =\\frac{25}{24}, \\, \\sec \\theta =\\frac{25}{7}, \\, \\cot \\theta =\\frac{7}{24}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>[latex]P\\left(\\frac{-15}{17},y\\right), \\, y<0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>[latex]P\\left(x,\\frac{\\sqrt{7}}{3}\\right), \\, x<0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572482140\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572482140\"]\r\n

a. [latex]x=\\frac{\u2212\\sqrt{2}}{3}[\/latex] b. [latex]\\sin \\theta =\\frac{\\sqrt{7}}{3}, \\, \\cos \\theta =\\frac{\u2212\\sqrt{2}}{3}, \\, \\tan \\theta =\\frac{\u2212\\sqrt{14}}{2}, \\, \\csc \\theta =\\frac{3\\sqrt{7}}{7}, \\, \\sec \\theta =\\frac{-3\\sqrt{2}}{2}, \\, \\cot \\theta =\\frac{\u2212\\sqrt{14}}{7}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26.\u00a0<\/strong>[latex]P\\left(x,\\frac{\u2212\\sqrt{15}}{4}\\right), \\, x>0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (27-34), simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.<\/p>\r\n\r\n

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27.\u00a0<\/strong>[latex]\\tan^2 x+\\sin x\\csc x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572480317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480317\"]\r\n

[latex]\\sec^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>[latex]\\sec x\\sin x\\cot x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]\\dfrac{\\tan^2 x}{\\sec^2 x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572449360\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572449360\"]\r\n

[latex]\\sin^2 x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30.\u00a0<\/strong>[latex]\\sec x-\\cos x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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31.\u00a0<\/strong>[latex](1+\\tan \\theta)^2-2\\tan \\theta [\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572552234\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572552234\"]\r\n

[latex]\\sec^2 \\theta [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]\\sin x(\\csc x-\\sin x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.\u00a0<\/strong>[latex]\\dfrac{\\cos t}{\\sin t} + \\dfrac{\\sin t}{1+\\cos t}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572169742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572169742\"]\r\n

[latex]\\large\\frac{1}{\\sin t} \\normalsize =\\csc t[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>[latex]\\dfrac{1+\\tan^2 \\alpha}{1+\\cot^2 \\alpha}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (35-42), verify that each equation is an identity.<\/p>\r\n\r\n

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35.\u00a0<\/strong>[latex]\\dfrac{\\tan \\theta \\cot \\theta}{\\csc \\theta} =\\sin \\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>[latex]\\dfrac{\\sec^2 \\theta}{\\tan \\theta} =\\sec \\theta \\csc \\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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37.\u00a0<\/strong>[latex]\\dfrac{\\sin t}{\\csc t} + \\dfrac{\\cos t}{\\sec t} =1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>[latex]\\dfrac{\\sin x}{\\cos x+1} + \\dfrac{\\cos x-1}{\\sin x} =0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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39.\u00a0<\/strong>[latex]\\cot \\gamma + \\tan \\gamma = \\sec \\gamma \\csc \\gamma [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>[latex]\\sin^2 \\beta + \\tan^2 \\beta + \\cos^2 \\beta = \\sec^2 \\beta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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41.\u00a0<\/strong>[latex]\\dfrac{1}{1-\\sin \\alpha} + \\dfrac{1}{1+\\sin \\alpha } =2\\sec^2 \\alpha [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42.\u00a0<\/strong>[latex]\\dfrac{\\tan \\theta -\\cot \\theta}{\\sin \\theta \\cos \\theta} =\\sec^2 \\theta -\\csc^2 \\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (43-50), solve the trigonometric equations on the interval [latex]0\\le \\theta <2\\pi[\/latex].<\/p>\r\n\r\n

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43.\u00a0<\/strong>[latex]2\\sin \\theta -1=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572480765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572480765\"]\r\n

[latex]\\theta = \\{\\frac{\\pi}{6}, \\, \\frac{5\\pi}{6}\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0<\/strong>[latex]1+\\cos \\theta =\\dfrac{1}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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45.\u00a0<\/strong>[latex]2\\tan^2 \\theta =2[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572141283\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572141283\"]\r\n

[latex]\\theta = \\{\\frac{\\pi}{4}, \\, \\frac{3\\pi}{4}, \\, \\frac{5\\pi}{4}, \\, \\frac{7\\pi}{4}\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0<\/strong>[latex]4\\sin^2 \\theta -2=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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47.\u00a0<\/strong>[latex]\\sqrt{3}\\cot \\theta +1=0[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572141745\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572141745\"]\r\n

[latex]\\theta = \\{\\frac{2\\pi}{3}, \\, \\frac{5\\pi}{3}\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>[latex]3\\sec \\theta -2\\sqrt{3}=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49.\u00a0<\/strong>[latex]2\\cos \\theta \\sin \\theta =\\sin \\theta [\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572210569\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572210569\"]\r\n

[latex]\\theta = \\{0, \\, \\pi, \\, \\frac{\\pi}{3}, \\, \\frac{5\\pi}{3}\\}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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50.\u00a0<\/strong>[latex]\\csc^2 \\theta +2\\csc \\theta +1=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (51-54), each graph is of the form [latex]y=A\\sin Bx[\/latex]\u00a0 or\u00a0 [latex]y=A\\cos Bx[\/latex], where [latex]B>0[\/latex]. Write the equation of the graph.<\/p>\r\n\r\n

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51.\r\n<\/strong>\"An<\/span>\r\n[reveal-answer q=\"fs-id1170572550732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572550732\"]\r\n

[latex]y=4\\sin\\Big(\\large\\frac{\\pi}{4} \\normalsize x\\Big)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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52.\r\n<\/strong>\"An<\/span><\/div>\r\n<\/div>\r\n
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53.\r\n<\/strong>\"An<\/span>\r\n[reveal-answer q=\"fs-id1170572173778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572173778\"]\r\n

[latex]y=\\cos(2\\pi x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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54.\r\n<\/strong>\"An<\/span><\/div>\r\n<\/div>\r\n

For the following exercises (55-60), find (a)<\/strong> the amplitude,\u00a0(b)<\/strong> the period, and\u00a0(c)<\/strong> the phase shift with direction for each function.<\/p>\r\n\r\n

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55.\u00a0<\/strong>[latex]y=\\sin\\left(x-\\dfrac{\\pi}{4}\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572481061\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572481061\"]\r\n

a. 1 b. [latex]2\\pi [\/latex] c. [latex]\\frac{\\pi}{4}[\/latex] units to the right<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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56.\u00a0<\/strong>[latex]y=3\\cos(2x+3)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57.\u00a0<\/strong>[latex]y=\\frac{-1}{2}\\sin\\left(\\frac{1}{4}x\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572242014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572242014\"]\r\n

a. [latex]\\frac{1}{2}[\/latex] b. [latex]8\\pi [\/latex] c. No phase shift<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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58.\u00a0<\/strong>[latex]y=2\\cos\\left(x-\\dfrac{\\pi}{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59.\u00a0<\/strong>[latex]y=-3\\sin(\\pi x+2)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572552210\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572552210\"]\r\n

a. 3 b. 2 c. [latex]\\frac{2}{\\pi}[\/latex] units to the left<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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60.\u00a0<\/strong>[latex]y=4\\cos\\left(2x-\\dfrac{\\pi}{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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61. [T]<\/strong> The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of [latex]120^{\\circ}[\/latex], how many inches does it move? Approximate to the nearest whole inch.<\/p>\r\n[reveal-answer q=\"fs-id1170572481438\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572481438\"]\r\n

Approximately 42 in.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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62. [T]<\/strong> Find the length of the arc intercepted by central angle [latex]\\theta [\/latex] in a circle of radius [latex]r[\/latex]. Round to the nearest hundredth.<\/p>\r\n

a. [latex]r=12.8[\/latex] cm, [latex]\\theta =\\frac{5\\pi}{6}[\/latex] rad\r\nb. [latex]r=4.378[\/latex] cm, [latex]\\theta =\\frac{7\\pi}{6}[\/latex] rad\r\nc. [latex]r=0.964[\/latex] cm, [latex]\\theta =50^{\\circ}[\/latex]\r\nd. [latex]r=8.55[\/latex] cm, [latex]\\theta =325^{\\circ}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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63. [T]<\/strong> As a point [latex]P[\/latex] moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed<\/em>, [latex]\\omega[\/latex], and is given by [latex]\\omega =\\dfrac{\\theta}{t}[\/latex], where [latex]\\theta [\/latex] is in radians and [latex]t[\/latex] is time. Find the angular speed for the given data. Round to the nearest thousandth.<\/p>\r\n

a. [latex]\\theta =\\frac{7\\pi}{4}[\/latex] rad, [latex]t=10[\/latex] sec\r\nb. [latex]\\theta =\\frac{3\\pi }{5}[\/latex] rad, [latex]t=8[\/latex] sec\r\nc. [latex]\\theta =\\frac{2\\pi }{9}[\/latex] rad, [latex]t=1[\/latex] min\r\nd. [latex]\\theta =23.76[\/latex] rad, [latex]t=14[\/latex] min<\/p>\r\n[reveal-answer q=\"fs-id1170572553987\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572553987\"]\r\n

a. 0.550 rad\/sec b. 0.236 rad\/sec c. 0.698 rad\/min d. 1.697 rad\/min<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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64. [T]<\/strong> A total of 250,000 m2<\/sup> of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.<\/p>\r\n\r\n

    \r\n \t
  1. Find the radius of the circular land area.<\/li>\r\n \t
  2. If the land area is to form a 45\u00b0 sector of a circle instead of a whole circle, find the length of the curved side.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
    \r\n
    \r\n

    65. [T]<\/strong> The area of an isosceles triangle with equal sides of length [latex]x[\/latex] is<\/p>\r\n

    [latex]\\frac{1}{2}x^2 \\sin \\theta[\/latex],<\/p>\r\n

    where [latex]\\theta [\/latex] is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle [latex]\\theta =\\frac{5\\pi}{12}[\/latex] rad.<\/p>\r\n[reveal-answer q=\"fs-id1170572173723\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572173723\"]\r\n

    [latex]\\approx 30.9 \\, \\text{in}^2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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    66. [T]<\/strong> A particle travels in a circular path at a constant angular speed [latex]\\omega[\/latex]. The angular speed is modeled by the function [latex]\\omega =9|\\cos(\\pi t-\\frac{\\pi}{12})|[\/latex]. Determine the angular speed at [latex]t=9[\/latex] sec.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

    \r\n
    \r\n

    67. [T]<\/strong> An alternating current for outlets in a home has voltage given by the function<\/p>\r\n

    [latex]V(t)=150\\cos 368t[\/latex],<\/p>\r\n

    where [latex]V[\/latex] is the voltage in volts at time [latex]t[\/latex] in seconds.<\/p>\r\n\r\n

      \r\n \t
    1. Find the period of the function and interpret its meaning.<\/li>\r\n \t
    2. Determine the number of periods that occur when 1 sec has passed.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572454371\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572454371\"]\r\n

      a. [latex]\\pi \/184[\/latex]; the voltage repeats every [latex]\\pi \/184[\/latex] sec b. Approximately 59 periods<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

      \r\n
      \r\n

      68. [T]<\/strong> The number of hours of daylight in a northeast city is modeled by the function<\/p>\r\n\r\n

      [latex]N(t)=12+3\\sin\\left[\\dfrac{2\\pi}{365}(t-79)\\right][\/latex],<\/div>\r\n

      where [latex]t[\/latex] is the number of days after January 1.<\/p>\r\n\r\n

        \r\n \t
      1. Find the amplitude and period.<\/li>\r\n \t
      2. Determine the number of hours of daylight on the longest day of the year.<\/li>\r\n \t
      3. Determine the number of hours of daylight on the shortest day of the year.<\/li>\r\n \t
      4. Determine the number of hours of daylight 90 days after January 1.<\/li>\r\n \t
      5. Sketch the graph of the function for one period starting on January 1.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
        \r\n
        \r\n

        69. [T]<\/strong> Suppose that [latex]T=50+10\\sin\\left[\\frac{\\pi}{12}(t-8)\\right][\/latex] is a mathematical model of the temperature (in degrees Fahrenheit) at [latex]t[\/latex] hours after midnight on a certain day of the week.<\/p>\r\n\r\n

          \r\n \t
        1. Determine the amplitude and period.<\/li>\r\n \t
        2. Find the temperature 7 hours after midnight.<\/li>\r\n \t
        3. At what time does [latex]T=60^{\\circ}[\/latex]?<\/li>\r\n \t
        4. Sketch the graph of [latex]T[\/latex] over [latex]0\\le t\\le 24[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"fs-id1170572222958\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572222958\"]\r\n

          a. Amplitude = 10; period = 24 b. [latex]47.4^{\\circ} F[\/latex] c. 14 hours later, or 2 p.m. d.<\/p>\r\n\"An<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

          \r\n
          \r\n

          70. [T]<\/strong> The function [latex]H(t)=8\\sin\\left(\\frac{\\pi}{6}t\\right)[\/latex] models the height [latex]H[\/latex] (in feet) of the tide [latex]t[\/latex] hours after midnight. Assume that [latex]t=0[\/latex] is midnight.<\/p>\r\n\r\n

            \r\n \t
          1. Find the amplitude and period.<\/li>\r\n \t
          2. Graph the function over one period.<\/li>\r\n \t
          3. What is the height of the tide at 4:30 a.m.?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"

            For the following exercises (1-5), convert each angle in degrees to radians. Write the answer as a multiple of [latex]\\pi[\/latex].<\/p>\n

            \n
            \n

            1.\u00a0<\/strong>240\u00b0<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\frac{4\\pi}{3}[\/latex] rad<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            2.\u00a0<\/strong>15\u00b0<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            3.\u00a0<\/strong>-60\u00b0<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\frac{\u2212\\pi }{3}[\/latex] rad<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            4.\u00a0<\/strong>-225\u00b0<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            5.\u00a0<\/strong>330\u00b0<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\frac{11\\pi}{6}[\/latex] rad<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            For the following exercises (6-10), convert each angle in radians to degrees.<\/p>\n

            \n
            \n

            6.\u00a0<\/strong>[latex]\\large \\frac{\\pi}{2}[\/latex] rad<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            7.\u00a0<\/strong>[latex]\\large \\frac{7\\pi}{6}[\/latex] rad<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            210\u00b0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            8.\u00a0<\/strong>[latex]\\large\\frac{11\\pi}{2}[\/latex] rad<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            9.\u00a0<\/strong>[latex]-3\\pi[\/latex] rad<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            -540\u00b0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            10.\u00a0<\/strong>[latex]\\large\\frac{5\\pi}{12}[\/latex] rad<\/p>\n<\/div>\n<\/div>\n

            Evaluate the following functional values (11-16).<\/p>\n

            \n
            \n

            11.\u00a0<\/strong>[latex]\\cos \\Big(\\large\\frac{4\\pi}{3}\\Big)[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            -1\/2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            12.\u00a0<\/strong>[latex]\\tan \\Big(\\large\\frac{19\\pi}{4}\\Big)[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            13.\u00a0<\/strong>[latex]\\sin\\left(-\\large\\frac{3\\pi}{4}\\right)[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]-\\large\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            14.\u00a0<\/strong>[latex]\\sec\\Big(\\large\\frac{\\pi}{6}\\Big)[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            15.\u00a0<\/strong>[latex]\\sin\\Big(\\large\\frac{\\pi}{12}\\Big)[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\large \\frac{\\sqrt{3}-1}{2\\sqrt{2}} \\normalsize = \\large \\frac{\\sqrt{6}-\\sqrt{2}}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            16.\u00a0<\/strong>[latex]\\cos \\Big(\\large \\frac{5\\pi}{12}\\Big)[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (17-22), consider triangle [latex]ABC[\/latex], a right triangle with a right angle at [latex]C[\/latex].\u00a0(a)<\/strong> Find the missing side of the triangle, and\u00a0(b)<\/strong>\u00a0find the six trigonometric function values for the angle at [latex]A[\/latex]. Where necessary, round to one decimal place.<\/p>\n

            \"An<\/span><\/p>\n

            \n
            \n

            17.\u00a0<\/strong>[latex]a=4, \\, c=7[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. [latex]b=5.7[\/latex] b. [latex]\\sin A=\\frac{4}{7}, \\, \\cos A=\\frac{5.7}{7}, \\, \\tan A=\\frac{4}{5.7}, \\, \\csc A=\\frac{7}{4}, \\, \\sec A=\\frac{7}{5.7}, \\, \\cot A=\\frac{5.7}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            18.\u00a0<\/strong>[latex]a=21, \\, c=29[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            19.\u00a0<\/strong>[latex]a=85.3, \\, b=125.5[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. [latex]c=151.7[\/latex] b. [latex]\\sin A=0.5623, \\, \\cos A=0.8273, \\, \\tan A=0.6797, \\, \\csc A=1.778, \\, \\sec A=1.209, \\, \\cot A=1.471[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            20.\u00a0<\/strong>[latex]b=40, \\, c=41[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            21.\u00a0<\/strong>[latex]a=84, \\, b=13[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. [latex]c=85[\/latex] b. [latex]\\sin A=\\frac{84}{85}, \\, \\cos A=\\frac{13}{85}, \\, \\tan A=\\frac{84}{13}, \\, \\csc A=\\frac{85}{84}, \\, \\sec A=\\frac{85}{13}, \\, \\cot A=\\frac{13}{84}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            22.\u00a0<\/strong>[latex]b=28, \\, c=35[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (23-26), [latex]P[\/latex] is a point on the unit circle. (a)<\/strong> Find the (exact) missing coordinate value of each point and (b)<\/strong> find the values of the six trigonometric functions for the angle [latex]\\theta[\/latex] with a terminal side that passes through point [latex]P[\/latex]. Rationalize all denominators.<\/p>\n

            \n
            \n

            23.\u00a0<\/strong>[latex]P\\left(\\frac{7}{25},y\\right), \\, y>0[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. [latex]y=\\frac{24}{25}[\/latex] b. [latex]\\sin \\theta =\\frac{24}{25}, \\, \\cos \\theta =\\frac{7}{25}, \\, \\tan \\theta =\\frac{24}{7}, \\, \\csc \\theta =\\frac{25}{24}, \\, \\sec \\theta =\\frac{25}{7}, \\, \\cot \\theta =\\frac{7}{24}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            24.\u00a0<\/strong>[latex]P\\left(\\frac{-15}{17},y\\right), \\, y<0[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            25.\u00a0<\/strong>[latex]P\\left(x,\\frac{\\sqrt{7}}{3}\\right), \\, x<0[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. [latex]x=\\frac{\u2212\\sqrt{2}}{3}[\/latex] b. [latex]\\sin \\theta =\\frac{\\sqrt{7}}{3}, \\, \\cos \\theta =\\frac{\u2212\\sqrt{2}}{3}, \\, \\tan \\theta =\\frac{\u2212\\sqrt{14}}{2}, \\, \\csc \\theta =\\frac{3\\sqrt{7}}{7}, \\, \\sec \\theta =\\frac{-3\\sqrt{2}}{2}, \\, \\cot \\theta =\\frac{\u2212\\sqrt{14}}{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            26.\u00a0<\/strong>[latex]P\\left(x,\\frac{\u2212\\sqrt{15}}{4}\\right), \\, x>0[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (27-34), simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only.<\/p>\n

            \n
            \n

            27.\u00a0<\/strong>[latex]\\tan^2 x+\\sin x\\csc x[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\sec^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            28.\u00a0<\/strong>[latex]\\sec x\\sin x\\cot x[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            29.\u00a0<\/strong>[latex]\\dfrac{\\tan^2 x}{\\sec^2 x}[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\sin^2 x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            30.\u00a0<\/strong>[latex]\\sec x-\\cos x[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            31.\u00a0<\/strong>[latex](1+\\tan \\theta)^2-2\\tan \\theta[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\sec^2 \\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            32.\u00a0<\/strong>[latex]\\sin x(\\csc x-\\sin x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            33.\u00a0<\/strong>[latex]\\dfrac{\\cos t}{\\sin t} + \\dfrac{\\sin t}{1+\\cos t}[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\large\\frac{1}{\\sin t} \\normalsize =\\csc t[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            34.\u00a0<\/strong>[latex]\\dfrac{1+\\tan^2 \\alpha}{1+\\cot^2 \\alpha}[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (35-42), verify that each equation is an identity.<\/p>\n

            \n
            \n

            35.\u00a0<\/strong>[latex]\\dfrac{\\tan \\theta \\cot \\theta}{\\csc \\theta} =\\sin \\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            36.\u00a0<\/strong>[latex]\\dfrac{\\sec^2 \\theta}{\\tan \\theta} =\\sec \\theta \\csc \\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            37.\u00a0<\/strong>[latex]\\dfrac{\\sin t}{\\csc t} + \\dfrac{\\cos t}{\\sec t} =1[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            38.\u00a0<\/strong>[latex]\\dfrac{\\sin x}{\\cos x+1} + \\dfrac{\\cos x-1}{\\sin x} =0[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            39.\u00a0<\/strong>[latex]\\cot \\gamma + \\tan \\gamma = \\sec \\gamma \\csc \\gamma[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            40.\u00a0<\/strong>[latex]\\sin^2 \\beta + \\tan^2 \\beta + \\cos^2 \\beta = \\sec^2 \\beta[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            41.\u00a0<\/strong>[latex]\\dfrac{1}{1-\\sin \\alpha} + \\dfrac{1}{1+\\sin \\alpha } =2\\sec^2 \\alpha[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            42.\u00a0<\/strong>[latex]\\dfrac{\\tan \\theta -\\cot \\theta}{\\sin \\theta \\cos \\theta} =\\sec^2 \\theta -\\csc^2 \\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (43-50), solve the trigonometric equations on the interval [latex]0\\le \\theta <2\\pi[\/latex].<\/p>\n

            \n
            \n

            43.\u00a0<\/strong>[latex]2\\sin \\theta -1=0[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\theta = \\{\\frac{\\pi}{6}, \\, \\frac{5\\pi}{6}\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            44.\u00a0<\/strong>[latex]1+\\cos \\theta =\\dfrac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            45.\u00a0<\/strong>[latex]2\\tan^2 \\theta =2[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\theta = \\{\\frac{\\pi}{4}, \\, \\frac{3\\pi}{4}, \\, \\frac{5\\pi}{4}, \\, \\frac{7\\pi}{4}\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            46.\u00a0<\/strong>[latex]4\\sin^2 \\theta -2=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            47.\u00a0<\/strong>[latex]\\sqrt{3}\\cot \\theta +1=0[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\theta = \\{\\frac{2\\pi}{3}, \\, \\frac{5\\pi}{3}\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            48.\u00a0<\/strong>[latex]3\\sec \\theta -2\\sqrt{3}=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

            \n
            \n

            49.\u00a0<\/strong>[latex]2\\cos \\theta \\sin \\theta =\\sin \\theta[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            [latex]\\theta = \\{0, \\, \\pi, \\, \\frac{\\pi}{3}, \\, \\frac{5\\pi}{3}\\}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            50.\u00a0<\/strong>[latex]\\csc^2 \\theta +2\\csc \\theta +1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n

            For the following exercises (51-54), each graph is of the form [latex]y=A\\sin Bx[\/latex]\u00a0 or\u00a0 [latex]y=A\\cos Bx[\/latex], where [latex]B>0[\/latex]. Write the equation of the graph.<\/p>\n

            \n
            51.
            \n<\/strong>\"An<\/span><\/p>\n
            Show Solution<\/span><\/p>\n
            \n

            [latex]y=4\\sin\\Big(\\large\\frac{\\pi}{4} \\normalsize x\\Big)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            52.
            \n<\/strong>\"An<\/span><\/div>\n<\/div>\n
            \n
            53.
            \n<\/strong>\"An<\/span><\/p>\n
            Show Solution<\/span><\/p>\n
            \n

            [latex]y=\\cos(2\\pi x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            54.
            \n<\/strong>\"An<\/span><\/div>\n<\/div>\n

            For the following exercises (55-60), find (a)<\/strong> the amplitude,\u00a0(b)<\/strong> the period, and\u00a0(c)<\/strong> the phase shift with direction for each function.<\/p>\n

            \n
            \n

            55.\u00a0<\/strong>[latex]y=\\sin\\left(x-\\dfrac{\\pi}{4}\\right)[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. 1 b. [latex]2\\pi[\/latex] c. [latex]\\frac{\\pi}{4}[\/latex] units to the right<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

            \n
            \n

            56.\u00a0<\/strong>[latex]y=3\\cos(2x+3)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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            57.\u00a0<\/strong>[latex]y=\\frac{-1}{2}\\sin\\left(\\frac{1}{4}x\\right)[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
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            a. [latex]\\frac{1}{2}[\/latex] b. [latex]8\\pi[\/latex] c. No phase shift<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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            58.\u00a0<\/strong>[latex]y=2\\cos\\left(x-\\dfrac{\\pi}{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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            59.\u00a0<\/strong>[latex]y=-3\\sin(\\pi x+2)[\/latex]<\/p>\n

            Show Solution<\/span><\/p>\n
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            a. 3 b. 2 c. [latex]\\frac{2}{\\pi}[\/latex] units to the left<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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            60.\u00a0<\/strong>[latex]y=4\\cos\\left(2x-\\dfrac{\\pi}{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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            61. [T]<\/strong> The diameter of a wheel rolling on the ground is 40 in. If the wheel rotates through an angle of [latex]120^{\\circ}[\/latex], how many inches does it move? Approximate to the nearest whole inch.<\/p>\n

            Show Solution<\/span><\/p>\n
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            Approximately 42 in.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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            62. [T]<\/strong> Find the length of the arc intercepted by central angle [latex]\\theta[\/latex] in a circle of radius [latex]r[\/latex]. Round to the nearest hundredth.<\/p>\n

            a. [latex]r=12.8[\/latex] cm, [latex]\\theta =\\frac{5\\pi}{6}[\/latex] rad
            \nb. [latex]r=4.378[\/latex] cm, [latex]\\theta =\\frac{7\\pi}{6}[\/latex] rad
            \nc. [latex]r=0.964[\/latex] cm, [latex]\\theta =50^{\\circ}[\/latex]
            \nd. [latex]r=8.55[\/latex] cm, [latex]\\theta =325^{\\circ}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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            63. [T]<\/strong> As a point [latex]P[\/latex] moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed<\/em>, [latex]\\omega[\/latex], and is given by [latex]\\omega =\\dfrac{\\theta}{t}[\/latex], where [latex]\\theta[\/latex] is in radians and [latex]t[\/latex] is time. Find the angular speed for the given data. Round to the nearest thousandth.<\/p>\n

            a. [latex]\\theta =\\frac{7\\pi}{4}[\/latex] rad, [latex]t=10[\/latex] sec
            \nb. [latex]\\theta =\\frac{3\\pi }{5}[\/latex] rad, [latex]t=8[\/latex] sec
            \nc. [latex]\\theta =\\frac{2\\pi }{9}[\/latex] rad, [latex]t=1[\/latex] min
            \nd. [latex]\\theta =23.76[\/latex] rad, [latex]t=14[\/latex] min<\/p>\n

            Show Solution<\/span><\/p>\n
            \n

            a. 0.550 rad\/sec b. 0.236 rad\/sec c. 0.698 rad\/min d. 1.697 rad\/min<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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            64. [T]<\/strong> A total of 250,000 m2<\/sup> of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.<\/p>\n

              \n
            1. Find the radius of the circular land area.<\/li>\n
            2. If the land area is to form a 45\u00b0 sector of a circle instead of a whole circle, find the length of the curved side.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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              65. [T]<\/strong> The area of an isosceles triangle with equal sides of length [latex]x[\/latex] is<\/p>\n

              [latex]\\frac{1}{2}x^2 \\sin \\theta[\/latex],<\/p>\n

              where [latex]\\theta[\/latex] is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle [latex]\\theta =\\frac{5\\pi}{12}[\/latex] rad.<\/p>\n

              Show Solution<\/span><\/p>\n
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              [latex]\\approx 30.9 \\, \\text{in}^2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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              66. [T]<\/strong> A particle travels in a circular path at a constant angular speed [latex]\\omega[\/latex]. The angular speed is modeled by the function [latex]\\omega =9|\\cos(\\pi t-\\frac{\\pi}{12})|[\/latex]. Determine the angular speed at [latex]t=9[\/latex] sec.<\/p>\n<\/div>\n<\/div>\n

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              67. [T]<\/strong> An alternating current for outlets in a home has voltage given by the function<\/p>\n

              [latex]V(t)=150\\cos 368t[\/latex],<\/p>\n

              where [latex]V[\/latex] is the voltage in volts at time [latex]t[\/latex] in seconds.<\/p>\n

                \n
              1. Find the period of the function and interpret its meaning.<\/li>\n
              2. Determine the number of periods that occur when 1 sec has passed.<\/li>\n<\/ol>\n
                Show Solution<\/span><\/p>\n
                \n

                a. [latex]\\pi \/184[\/latex]; the voltage repeats every [latex]\\pi \/184[\/latex] sec b. Approximately 59 periods<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                68. [T]<\/strong> The number of hours of daylight in a northeast city is modeled by the function<\/p>\n

                [latex]N(t)=12+3\\sin\\left[\\dfrac{2\\pi}{365}(t-79)\\right][\/latex],<\/div>\n

                where [latex]t[\/latex] is the number of days after January 1.<\/p>\n

                  \n
                1. Find the amplitude and period.<\/li>\n
                2. Determine the number of hours of daylight on the longest day of the year.<\/li>\n
                3. Determine the number of hours of daylight on the shortest day of the year.<\/li>\n
                4. Determine the number of hours of daylight 90 days after January 1.<\/li>\n
                5. Sketch the graph of the function for one period starting on January 1.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
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                  69. [T]<\/strong> Suppose that [latex]T=50+10\\sin\\left[\\frac{\\pi}{12}(t-8)\\right][\/latex] is a mathematical model of the temperature (in degrees Fahrenheit) at [latex]t[\/latex] hours after midnight on a certain day of the week.<\/p>\n

                    \n
                  1. Determine the amplitude and period.<\/li>\n
                  2. Find the temperature 7 hours after midnight.<\/li>\n
                  3. At what time does [latex]T=60^{\\circ}[\/latex]?<\/li>\n
                  4. Sketch the graph of [latex]T[\/latex] over [latex]0\\le t\\le 24[\/latex].<\/li>\n<\/ol>\n
                    Show Solution<\/span><\/p>\n
                    \n

                    a. Amplitude = 10; period = 24 b. [latex]47.4^{\\circ} F[\/latex] c. 14 hours later, or 2 p.m. d.<\/p>\n

                    \"An<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n

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                    70. [T]<\/strong> The function [latex]H(t)=8\\sin\\left(\\frac{\\pi}{6}t\\right)[\/latex] models the height [latex]H[\/latex] (in feet) of the tide [latex]t[\/latex] hours after midnight. Assume that [latex]t=0[\/latex] is midnight.<\/p>\n

                      \n
                    1. Find the amplitude and period.<\/li>\n
                    2. Graph the function over one period.<\/li>\n
                    3. What is the height of the tide at 4:30 a.m.?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n\n\t\t\t
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