{"id":2213,"date":"2018-01-11T21:36:11","date_gmt":"2018-01-11T21:36:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-openstax-calculus1\/back-matter\/review-of-pre-calculus\/"},"modified":"2018-01-11T21:36:11","modified_gmt":"2018-01-11T21:36:11","slug":"review-of-pre-calculus","status":"publish","type":"back-matter","link":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/back-matter\/review-of-pre-calculus\/","title":{"raw":"Review of Pre-Calculus","rendered":"Review of Pre-Calculus"},"content":{"raw":"

Formulas from Geometry<\/h1>

[latex]A=\\text{area},[\/latex][latex]V=\\text{Volume},\\text{ and }[\/latex][latex]S=\\text{lateral surface area}[\/latex]<\/p>\"The<\/span>\"The<\/span><\/div>

Formulas from Algebra<\/h1>

Laws of Exponents<\/h2>

[latex]\\begin{array}{ccccccccccccc}\\hfill {x}^{m}{x}^{n}& =\\hfill & {x}^{m+n}\\hfill & & & \\hfill \\frac{{x}^{m}}{{x}^{n}}& =\\hfill & {x}^{m-n}\\hfill & & & \\hfill {({x}^{m})}^{n}& =\\hfill & {x}^{mn}\\hfill \\\\ \\hfill {x}^{\\text{\u2212}n}& =\\hfill & \\frac{1}{{x}^{n}}\\hfill & & & \\hfill {(xy)}^{n}& =\\hfill & {x}^{n}{y}^{n}\\hfill & & & \\hfill {(\\frac{x}{y})}^{n}& =\\hfill & \\frac{{x}^{n}}{{y}^{n}}\\hfill \\\\ \\hfill {x}^{1\\text{\/}n}& =\\hfill & \\sqrt[n]{x}\\hfill & & & \\hfill \\sqrt[n]{xy}& =\\hfill & \\sqrt[n]{x}\\sqrt[n]{y}\\hfill & & & \\hfill \\sqrt[n]{\\frac{x}{y}}& =\\hfill & \\frac{\\sqrt[n]{x}}{\\sqrt[n]{y}}\\hfill \\\\ \\hfill {x}^{m\\text{\/}n}& =\\hfill & \\sqrt[n]{{x}^{m}}={(\\sqrt[n]{x})}^{m}\\hfill & & & & & & & & & & \\end{array}[\/latex]<\/p><\/div>

Special Factorizations<\/h2>

[latex]\\begin{array}{ccc}\\hfill {x}^{2}-{y}^{2}& =\\hfill & (x+y)(x-y)\\hfill \\\\ \\hfill {x}^{3}+{y}^{3}& =\\hfill & (x+y)({x}^{2}-xy+{y}^{2})\\hfill \\\\ \\hfill {x}^{3}-{y}^{3}& =\\hfill & (x-y)({x}^{2}+xy+{y}^{2})\\hfill \\end{array}[\/latex]<\/p><\/div>

Quadratic Formula<\/h2>

If [latex]a{x}^{2}+bx+c=0,[\/latex] then [latex]x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ca}}{2a}.[\/latex]<\/p><\/div>

Binomial Theorem<\/h2>

[latex]{(a+b)}^{n}={a}^{n}+(\\begin{array}{l}n\\\\ 1\\end{array}){a}^{n-1}b+(\\begin{array}{l}n\\\\ 2\\end{array}){a}^{n-2}{b}^{2}+\\cdots +(\\begin{array}{c}n\\\\ n-1\\end{array})a{b}^{n-1}+{b}^{n},[\/latex]<\/p>

where [latex](\\begin{array}{l}n\\\\ k\\end{array})=\\frac{n(n-1)(n-2)\\cdots (n-k+1)}{k(k-1)(k-2)\\cdots 3\\cdot 2\\cdot 1}=\\frac{n!}{k!(n-k)!}[\/latex]<\/p><\/div><\/div>

Formulas from Trigonometry<\/h1>

Right-Angle Trigonometry<\/h2>

[latex]\\begin{array}{cccc} \\sin \\theta =\\frac{\\text{opp}}{\\text{hyp}}\\hfill & & & \\csc \\theta =\\frac{\\text{hyp}}{\\text{opp}}\\hfill \\\\ \\cos \\theta =\\frac{\\text{adj}}{\\text{hyp}}\\hfill & & & \\sec \\theta =\\frac{\\text{hyp}}{\\text{adj}}\\hfill \\\\ \\tan \\theta =\\frac{\\text{opp}}{\\text{adj}}\\hfill & & & \\cot \\theta =\\frac{\\text{adj}}{\\text{opp}}\\hfill \\end{array}[\/latex]<\/p>\"The<\/span><\/div>

Trigonometric Functions of Important Angles<\/h2>
[latex]\\theta [\/latex]<\/td>[latex]\\text{Radians}[\/latex]<\/td>[latex] \\sin \\theta [\/latex]<\/td>[latex] \\cos \\theta [\/latex]<\/td>[latex] \\tan \\theta [\/latex]<\/td><\/tr>
0\u00b0<\/td>0<\/td>0<\/td>1<\/td>0<\/td><\/tr>
30\u00b0<\/td>[latex]\\text{\u03c0}\\text{\/}\\text{6}[\/latex]<\/td>[latex]1\\text{\/}2[\/latex]<\/td>[latex]\\sqrt{3}\\text{\/}2[\/latex]<\/td>[latex]\\sqrt{3}\\text{\/}3[\/latex]<\/td><\/tr>
45\u00b0<\/td>[latex]\\text{\u03c0}\\text{\/}\\text{4}[\/latex]<\/td>[latex]\\sqrt{2}\\text{\/}2[\/latex]<\/td>[latex]\\sqrt{2}\\text{\/}2[\/latex]<\/td>1<\/td><\/tr>
60\u00b0<\/td>[latex]\\text{\u03c0}\\text{\/}\\text{3}[\/latex]<\/td>[latex]\\sqrt{3}\\text{\/}2[\/latex]<\/td>[latex]1\\text{\/}2[\/latex]<\/td>[latex]\\sqrt{3}[\/latex]<\/td><\/tr>
90\u00b0<\/td>[latex]\\text{\u03c0}\\text{\/}2[\/latex]<\/td>1<\/td>0<\/td>\u2014<\/td><\/tr><\/tbody><\/table><\/div>

Fundamental Identities<\/h2>

[latex]\\begin{array}{cccccccc}\\hfill { \\sin }^{2}\\theta +{ \\cos }^{2}\\theta & =\\hfill & 1\\hfill & & & \\hfill \\sin (\\text{\u2212}\\theta )& =\\hfill & \\text{\u2212} \\sin \\theta \\hfill \\\\ \\hfill 1+{ \\tan }^{2}\\theta & =\\hfill & { \\sec }^{2}\\theta \\hfill & & & \\hfill \\cos (\\text{\u2212}\\theta )& =\\hfill & \\cos \\theta \\hfill \\\\ \\hfill 1+{ \\cot }^{2}\\theta & =\\hfill & { \\csc }^{2}\\theta \\hfill & & & \\hfill \\tan (\\text{\u2212}\\theta )& =\\hfill & \\text{\u2212} \\tan \\theta \\hfill \\\\ \\hfill \\sin (\\frac{\\pi }{2}-\\theta )& =\\hfill & \\cos \\theta \\hfill & & & \\hfill \\sin (\\theta +2\\pi )& =\\hfill & \\sin \\theta \\hfill \\\\ \\hfill \\cos (\\frac{\\pi }{2}-\\theta )& =\\hfill & \\sin \\theta \\hfill & & & \\hfill \\cos (\\theta +2\\pi )& =\\hfill & \\cos \\theta \\hfill \\\\ \\hfill \\tan (\\frac{\\pi }{2}-\\theta )& =\\hfill & \\cot \\theta \\hfill & & & \\hfill \\tan (\\theta +\\pi )& =\\hfill & \\tan \\theta \\hfill \\end{array}[\/latex]<\/p><\/div>

Law of Sines<\/h2>

[latex]\\frac{ \\sin A}{a}=\\frac{ \\sin B}{b}=\\frac{ \\sin C}{c}[\/latex]<\/p>\"The<\/span><\/div>

Law of Cosines<\/h2>

[latex]\\begin{array}{ccc}\\hfill {a}^{2}& =\\hfill & {b}^{2}+{c}^{2}-2bc \\cos A\\hfill \\\\ \\hfill {b}^{2}& =\\hfill & {a}^{2}+{c}^{2}-2ac \\cos B\\hfill \\\\ \\hfill {c}^{2}& =\\hfill & {a}^{2}+{b}^{2}-2ab \\cos C\\hfill \\end{array}[\/latex]<\/p><\/div>

Addition and Subtraction Formulas<\/h2>

[latex]\\begin{array}{ccc}\\hfill \\sin (x+y)& =\\hfill & \\sin x \\cos y+ \\cos x \\sin y\\hfill \\\\ \\hfill \\sin (x-y)& =\\hfill & \\sin x \\cos y- \\cos x \\sin y\\hfill \\\\ \\hfill \\cos (x+y)& =\\hfill & \\cos x \\cos y- \\sin x \\sin y\\hfill \\\\ \\hfill \\cos (x-y)& =\\hfill & \\cos x \\cos y+ \\sin x \\sin y\\hfill \\\\ \\hfill \\tan (x+y)& =\\hfill & \\frac{ \\tan x+ \\tan y}{1- \\tan x \\tan y}\\hfill \\\\ \\hfill \\tan (x-y)& =\\hfill & \\frac{ \\tan x- \\tan y}{1+ \\tan x \\tan y}\\hfill \\end{array}[\/latex]<\/p><\/div>

Double-Angle Formulas<\/h2>

[latex]\\begin{array}{ccc}\\hfill \\sin 2x& =\\hfill & 2 \\sin x \\cos x\\hfill \\\\ \\hfill \\cos 2x& =\\hfill & { \\cos }^{2}x-{ \\sin }^{2}x=2{ \\cos }^{2}x-1=1-2{ \\sin }^{2}x\\hfill \\\\ \\hfill \\tan 2x& =\\hfill & \\frac{2 \\tan x}{1-{ \\tan }^{2}x}\\hfill \\end{array}[\/latex]<\/p><\/div>

Half-Angle Formulas<\/h2>

[latex]\\begin{array}{ccc}\\hfill { \\sin }^{2}x& =\\hfill & \\frac{1- \\cos 2x}{2}\\hfill \\\\ \\hfill { \\cos }^{2}x& =\\hfill & \\frac{1+ \\cos 2x}{2}\\hfill \\end{array}[\/latex]<\/p><\/div><\/div>","rendered":"

\n

Formulas from Geometry<\/h1>\n

[latex]A=\\text{area},[\/latex][latex]V=\\text{Volume},\\text{ and }[\/latex][latex]S=\\text{lateral surface area}[\/latex]<\/p>\n

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

\n

Formulas from Algebra<\/h1>\n
\n

Laws of Exponents<\/h2>\n

[latex]\\begin{array}{ccccccccccccc}\\hfill {x}^{m}{x}^{n}& =\\hfill & {x}^{m+n}\\hfill & & & \\hfill \\frac{{x}^{m}}{{x}^{n}}& =\\hfill & {x}^{m-n}\\hfill & & & \\hfill {({x}^{m})}^{n}& =\\hfill & {x}^{mn}\\hfill \\\\ \\hfill {x}^{\\text{\u2212}n}& =\\hfill & \\frac{1}{{x}^{n}}\\hfill & & & \\hfill {(xy)}^{n}& =\\hfill & {x}^{n}{y}^{n}\\hfill & & & \\hfill {(\\frac{x}{y})}^{n}& =\\hfill & \\frac{{x}^{n}}{{y}^{n}}\\hfill \\\\ \\hfill {x}^{1\\text{\/}n}& =\\hfill & \\sqrt[n]{x}\\hfill & & & \\hfill \\sqrt[n]{xy}& =\\hfill & \\sqrt[n]{x}\\sqrt[n]{y}\\hfill & & & \\hfill \\sqrt[n]{\\frac{x}{y}}& =\\hfill & \\frac{\\sqrt[n]{x}}{\\sqrt[n]{y}}\\hfill \\\\ \\hfill {x}^{m\\text{\/}n}& =\\hfill & \\sqrt[n]{{x}^{m}}={(\\sqrt[n]{x})}^{m}\\hfill & & & & & & & & & & \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Special Factorizations<\/h2>\n

[latex]\\begin{array}{ccc}\\hfill {x}^{2}-{y}^{2}& =\\hfill & (x+y)(x-y)\\hfill \\\\ \\hfill {x}^{3}+{y}^{3}& =\\hfill & (x+y)({x}^{2}-xy+{y}^{2})\\hfill \\\\ \\hfill {x}^{3}-{y}^{3}& =\\hfill & (x-y)({x}^{2}+xy+{y}^{2})\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Quadratic Formula<\/h2>\n

If [latex]a{x}^{2}+bx+c=0,[\/latex] then [latex]x=\\frac{\\text{\u2212}b\u00b1\\sqrt{{b}^{2}-4ca}}{2a}.[\/latex]<\/p>\n<\/div>\n

\n

Binomial Theorem<\/h2>\n

[latex]{(a+b)}^{n}={a}^{n}+(\\begin{array}{l}n\\\\ 1\\end{array}){a}^{n-1}b+(\\begin{array}{l}n\\\\ 2\\end{array}){a}^{n-2}{b}^{2}+\\cdots +(\\begin{array}{c}n\\\\ n-1\\end{array})a{b}^{n-1}+{b}^{n},[\/latex]<\/p>\n

where [latex](\\begin{array}{l}n\\\\ k\\end{array})=\\frac{n(n-1)(n-2)\\cdots (n-k+1)}{k(k-1)(k-2)\\cdots 3\\cdot 2\\cdot 1}=\\frac{n!}{k!(n-k)!}[\/latex]<\/p>\n<\/div>\n<\/div>\n

\n

Formulas from Trigonometry<\/h1>\n
\n

Right-Angle Trigonometry<\/h2>\n

[latex]\\begin{array}{cccc} \\sin \\theta =\\frac{\\text{opp}}{\\text{hyp}}\\hfill & & & \\csc \\theta =\\frac{\\text{hyp}}{\\text{opp}}\\hfill \\\\ \\cos \\theta =\\frac{\\text{adj}}{\\text{hyp}}\\hfill & & & \\sec \\theta =\\frac{\\text{hyp}}{\\text{adj}}\\hfill \\\\ \\tan \\theta =\\frac{\\text{opp}}{\\text{adj}}\\hfill & & & \\cot \\theta =\\frac{\\text{adj}}{\\text{opp}}\\hfill \\end{array}[\/latex]<\/p>\n

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

\n

Trigonometric Functions of Important Angles<\/h2>\n\n\n\n\n\n\n\n\n
[latex]\\theta[\/latex]<\/td>\n[latex]\\text{Radians}[\/latex]<\/td>\n[latex]\\sin \\theta[\/latex]<\/td>\n[latex]\\cos \\theta[\/latex]<\/td>\n[latex]\\tan \\theta[\/latex]<\/td>\n<\/tr>\n
0\u00b0<\/td>\n0<\/td>\n0<\/td>\n1<\/td>\n0<\/td>\n<\/tr>\n
30\u00b0<\/td>\n[latex]\\text{\u03c0}\\text{\/}\\text{6}[\/latex]<\/td>\n[latex]1\\text{\/}2[\/latex]<\/td>\n[latex]\\sqrt{3}\\text{\/}2[\/latex]<\/td>\n[latex]\\sqrt{3}\\text{\/}3[\/latex]<\/td>\n<\/tr>\n
45\u00b0<\/td>\n[latex]\\text{\u03c0}\\text{\/}\\text{4}[\/latex]<\/td>\n[latex]\\sqrt{2}\\text{\/}2[\/latex]<\/td>\n[latex]\\sqrt{2}\\text{\/}2[\/latex]<\/td>\n1<\/td>\n<\/tr>\n
60\u00b0<\/td>\n[latex]\\text{\u03c0}\\text{\/}\\text{3}[\/latex]<\/td>\n[latex]\\sqrt{3}\\text{\/}2[\/latex]<\/td>\n[latex]1\\text{\/}2[\/latex]<\/td>\n[latex]\\sqrt{3}[\/latex]<\/td>\n<\/tr>\n
90\u00b0<\/td>\n[latex]\\text{\u03c0}\\text{\/}2[\/latex]<\/td>\n1<\/td>\n0<\/td>\n\u2014<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
\n

Fundamental Identities<\/h2>\n

[latex]\\begin{array}{cccccccc}\\hfill { \\sin }^{2}\\theta +{ \\cos }^{2}\\theta & =\\hfill & 1\\hfill & & & \\hfill \\sin (\\text{\u2212}\\theta )& =\\hfill & \\text{\u2212} \\sin \\theta \\hfill \\\\ \\hfill 1+{ \\tan }^{2}\\theta & =\\hfill & { \\sec }^{2}\\theta \\hfill & & & \\hfill \\cos (\\text{\u2212}\\theta )& =\\hfill & \\cos \\theta \\hfill \\\\ \\hfill 1+{ \\cot }^{2}\\theta & =\\hfill & { \\csc }^{2}\\theta \\hfill & & & \\hfill \\tan (\\text{\u2212}\\theta )& =\\hfill & \\text{\u2212} \\tan \\theta \\hfill \\\\ \\hfill \\sin (\\frac{\\pi }{2}-\\theta )& =\\hfill & \\cos \\theta \\hfill & & & \\hfill \\sin (\\theta +2\\pi )& =\\hfill & \\sin \\theta \\hfill \\\\ \\hfill \\cos (\\frac{\\pi }{2}-\\theta )& =\\hfill & \\sin \\theta \\hfill & & & \\hfill \\cos (\\theta +2\\pi )& =\\hfill & \\cos \\theta \\hfill \\\\ \\hfill \\tan (\\frac{\\pi }{2}-\\theta )& =\\hfill & \\cot \\theta \\hfill & & & \\hfill \\tan (\\theta +\\pi )& =\\hfill & \\tan \\theta \\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Law of Sines<\/h2>\n

[latex]\\frac{ \\sin A}{a}=\\frac{ \\sin B}{b}=\\frac{ \\sin C}{c}[\/latex]<\/p>\n

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

\n

Law of Cosines<\/h2>\n

[latex]\\begin{array}{ccc}\\hfill {a}^{2}& =\\hfill & {b}^{2}+{c}^{2}-2bc \\cos A\\hfill \\\\ \\hfill {b}^{2}& =\\hfill & {a}^{2}+{c}^{2}-2ac \\cos B\\hfill \\\\ \\hfill {c}^{2}& =\\hfill & {a}^{2}+{b}^{2}-2ab \\cos C\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Addition and Subtraction Formulas<\/h2>\n

[latex]\\begin{array}{ccc}\\hfill \\sin (x+y)& =\\hfill & \\sin x \\cos y+ \\cos x \\sin y\\hfill \\\\ \\hfill \\sin (x-y)& =\\hfill & \\sin x \\cos y- \\cos x \\sin y\\hfill \\\\ \\hfill \\cos (x+y)& =\\hfill & \\cos x \\cos y- \\sin x \\sin y\\hfill \\\\ \\hfill \\cos (x-y)& =\\hfill & \\cos x \\cos y+ \\sin x \\sin y\\hfill \\\\ \\hfill \\tan (x+y)& =\\hfill & \\frac{ \\tan x+ \\tan y}{1- \\tan x \\tan y}\\hfill \\\\ \\hfill \\tan (x-y)& =\\hfill & \\frac{ \\tan x- \\tan y}{1+ \\tan x \\tan y}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Double-Angle Formulas<\/h2>\n

[latex]\\begin{array}{ccc}\\hfill \\sin 2x& =\\hfill & 2 \\sin x \\cos x\\hfill \\\\ \\hfill \\cos 2x& =\\hfill & { \\cos }^{2}x-{ \\sin }^{2}x=2{ \\cos }^{2}x-1=1-2{ \\sin }^{2}x\\hfill \\\\ \\hfill \\tan 2x& =\\hfill & \\frac{2 \\tan x}{1-{ \\tan }^{2}x}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Half-Angle Formulas<\/h2>\n

[latex]\\begin{array}{ccc}\\hfill { \\sin }^{2}x& =\\hfill & \\frac{1- \\cos 2x}{2}\\hfill \\\\ \\hfill { \\cos }^{2}x& =\\hfill & \\frac{1+ \\cos 2x}{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":311,"menu_order":3,"template":"","meta":{"_candela_citation":"","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-2213","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter\/2213","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter\/2213\/revisions"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter\/2213\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/media?parent=2213"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/pressbooks\/v2\/back-matter-type?post=2213"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/contributor?post=2213"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-geneseo-openstax-calculus1-1\/wp-json\/wp\/v2\/license?post=2213"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}