Chapter Review Exercises
Graphs of the Sine and Cosine Functions
For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
[latex]f\left(x\right)=-3\mathrm{cos}\text{ }x+3[/latex]
![A graph of two periods of a function with a cosine parent function. The graph has a range of [0,6] graphed over -2pi to 2pi. Maximums as -pi and pi.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133928/CNX_Precalc_Figure_06_03_206.jpg)
[latex]f\left(x\right)=\frac{1}{4}\mathrm{sin}\text{ }x[/latex]
[latex]f\left(x\right)=3\mathrm{cos}\left(x+\frac{\pi }{6}\right)[/latex]
![A graph of four periods of a function with a cosine parent function. Graphed from -4pi to 4pi. Range is [-3,3].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133930/CNX_Precalc_Figure_06_03_208.jpg)
[latex]f\left(x\right)=-2\mathrm{sin}\left(x-\frac{2\pi }{3}\right)[/latex]
[latex]f\left(x\right)=3\mathrm{sin}\left(x-\frac{\pi }{4}\right)-4[/latex]
![A graph of two periods of a sinusoidal function. Range is [-7,-1]. Maximums at -5pi/4 and 3pi/4.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133933/CNX_Precalc_Figure_06_03_210.jpg)
[latex]f\left(x\right)=2\left(\mathrm{cos}\left(x-\frac{4\pi }{3}\right)+1\right)[/latex]
[latex]f\left(x\right)=6\mathrm{sin}\left(3x-\frac{\pi }{6}\right)-1[/latex]
![A sinusoidal graph over two periods. Range is [-7,5], amplitude is 6, and period is 2pi/3.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133936/CNX_Precalc_Figure_06_03_212.jpg)
[latex]f\left(x\right)=-100\mathrm{sin}\left(50x-20\right)[/latex]
Graphs of the Other Trigonometric Functions
For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
[latex]f\left(x\right)=\mathrm{tan}\text{ }x-4[/latex]
[latex]f\left(x\right)=2\mathrm{tan}\left(x-\frac{\pi }{6}\right)[/latex]
[latex]f\left(x\right)=-3\mathrm{tan}\left(4x\right)-2[/latex]
[latex]f\left(x\right)=0.2\mathrm{cos}\left(0.1x\right)+0.3[/latex]
For the following exercises, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.
[latex]f\left(x\right)=\frac{1}{3}\mathrm{sec}\text{ }x[/latex]
[latex]f\left(x\right)=3\mathrm{cot}\text{ }x[/latex]
[latex]f\left(x\right)=4\mathrm{csc}\left(5x\right)[/latex]
[latex]f\left(x\right)=8\mathrm{sec}\left(\frac{1}{4}x\right)[/latex]
[latex]f\left(x\right)=\frac{2}{3}\mathrm{csc}\left(\frac{1}{2}x\right)[/latex]
[latex]f\left(x\right)=-\mathrm{csc}\left(2x+\pi \right)[/latex]
For the following exercises, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function:[latex]\text{ }y=12,000+8,000\mathrm{sin}\left(0.628x\right),\text{ }[/latex]where the domain is the years since 1980 and the range is the population of the city.
What is the largest and smallest population the city may have?
Graph the function on the domain of[latex]\text{ }\left[0,40\right][/latex].
What are the amplitude, period, and phase shift for the function?
Over this domain, when does the population reach 18,000? 13,000?
What is the predicted population in 2007? 2010?
For the following exercises, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.
Suppose the graph of the displacement function is shown in (Figure), where the values on the x-axis represent the time in seconds and the y-axis represents the displacement in inches. Give the equation that models the vertical displacement of the weight on the spring.
![A graph of a consine function over one period. Graphed on the domain of [0,10]. Range is [-5,5].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133952/CNX_Precalc_Figure_06_03_225.jpg)
At time = 0, what is the displacement of the weight?
At what time does the displacement from the equilibrium point equal zero?
What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?
Inverse Trigonometric Functions
For the following exercises, find the exact value without the aid of a calculator.
[latex]{\mathrm{sin}}^{-1}\left(1\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)[/latex]
[latex]{\mathrm{tan}}^{-1}\left(-1\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(\frac{1}{\sqrt{2}}\right)[/latex]
[latex]{\mathrm{sin}}^{-1}\left(\frac{-\sqrt{3}}{2}\right)[/latex]
[latex]{\mathrm{sin}}^{-1}\left(\mathrm{cos}\left(\frac{\pi }{6}\right)\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(\mathrm{tan}\left(\frac{3\pi }{4}\right)\right)[/latex]
[latex]\mathrm{sin}\left({\mathrm{sec}}^{-1}\left(\frac{3}{5}\right)\right)[/latex]
[latex]\mathrm{cot}\left({\mathrm{sin}}^{-1}\left(\frac{3}{5}\right)\right)[/latex]
[latex]\mathrm{tan}\left({\mathrm{cos}}^{-1}\left(\frac{5}{13}\right)\right)[/latex]
[latex]\mathrm{sin}\left({\mathrm{cos}}^{-1}\left(\frac{x}{x+1}\right)\right)[/latex]
Graph[latex]\text{ }f\left(x\right)=\mathrm{cos}\text{ }x\text{ }[/latex]and[latex]\text{ }f\left(x\right)=\mathrm{sec}\text{ }x\text{ }[/latex]on the interval[latex]\text{ }\left[0,2\pi \right)\text{ }[/latex]and explain any observations.
Graph[latex]\text{ }f\left(x\right)=\mathrm{sin}\text{ }x\text{ }[/latex]and[latex]\text{ }f\left(x\right)=\mathrm{csc}\text{ }x\text{ }[/latex]and explain any observations.
Graph the function[latex]f\text{ }\left(x\right)=\frac{x}{1}-\frac{{x}^{3}}{3!}+\frac{{x}^{5}}{5!}-\frac{{x}^{7}}{7!}\text{ }[/latex]on the interval[latex]\text{ }\left[-1,1\right]\text{ }[/latex]and compare the graph to the graph of[latex]\text{ }f\left(x\right)=\mathrm{sin}\text{ }x\text{ }[/latex]on the same interval. Describe any observations.
![Two graphs of two identical functions on the interval [-1 to 1]. Both graphs appear sinusoidal.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07133958/CNX_Precalc_Figure_06_03_228.jpg)
Chapter Practice Test
For the following exercises, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.
[latex]f\left(x\right)=0.5\mathrm{sin}\text{ }x[/latex]
![A graph of two periods of a sinusoidal function, graphed over -2pi to 2pi. The range is [-0.5,0.5]. X-intercepts at multiples of pi.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134000/CNX_Precalc_Figure_06_03_229.jpg)
[latex]f\left(x\right)=5\mathrm{cos}\text{ }x[/latex]
[latex]f\left(x\right)=5\mathrm{sin}\text{ }x[/latex]
![Two periods of a sine function, graphed over -2pi to 2pi. The range is [-5,5], amplitude of 5, period of 2pi.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134003/CNX_Precalc_Figure_06_03_231.jpg)
[latex]f\left(x\right)=\mathrm{sin}\left(3x\right)[/latex]
[latex]f\left(x\right)=-\mathrm{cos}\left(x+\frac{\pi }{3}\right)+1[/latex]
![A graph of two periods of a cosine function, graphed over -7pi/3 to 5pi/3. Range is [0,2], Period is 2pi, amplitude is1.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134005/CNX_Precalc_Figure_06_03_233.jpg)
[latex]f\left(x\right)=5\mathrm{sin}\left(3\left(x-\frac{\pi }{6}\right)\right)+4[/latex]
[latex]f\left(x\right)=3\mathrm{cos}\left(\frac{1}{3}x-\frac{5\pi }{6}\right)[/latex]
![A graph of two periods of a cosine function, over -7pi/2 to 17pi/2. The range is [-3,3], period is 6pi, and amplitude is 3.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134008/CNX_Precalc_Figure_06_03_235.jpg)
[latex]f\left(x\right)=\mathrm{tan}\left(4x\right)[/latex]
[latex]f\left(x\right)=-2\mathrm{tan}\left(x-\frac{7\pi }{6}\right)+2[/latex]
[latex]f\left(x\right)=\pi \mathrm{cos}\left(3x+\pi \right)[/latex]
[latex]f\left(x\right)=5\mathrm{csc}\left(3x\right)[/latex]
[latex]f\left(x\right)=\pi \mathrm{sec}\left(\frac{\pi }{2}x\right)[/latex]
[latex]f\left(x\right)=2\mathrm{csc}\left(x+\frac{\pi }{4}\right)-3[/latex]
For the following exercises, determine the amplitude, period, and midline of the graph, and then find a formula for the function.
Give in terms of a sine function.
![A graph of two periods of a sine function, graphed from -2 to 2. Range is [-6,-2], period is 2, and amplitude is 2.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134020/CNX_Precalc_Figure_06_03_242.jpg)
Give in terms of a sine function.
![A graph of two periods of a sine function, graphed over -2 to 2. Range is [-2,2], period is 2, and amplitude is 2.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134022/CNX_Precalc_Figure_06_03_243.jpg)
Give in terms of a tangent function.
For the following exercises, find the amplitude, period, phase shift, and midline.
[latex]y=\mathrm{sin}\left(\frac{\pi }{6}x+\pi \right)-3[/latex]
[latex]y=8\mathrm{sin}\left(\frac{7\pi }{6}x+\frac{7\pi }{2}\right)+6[/latex]
The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming[latex]\text{ }t\text{ }[/latex]is the number of hours since midnight, find a function for the temperature,[latex]\text{ }D,\text{ }[/latex]in terms of[latex]\text{ }t.[/latex]
Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.
For the following exercises, find the period and horizontal shift of each function.
[latex]g\left(x\right)=3\mathrm{tan}\left(6x+42\right)[/latex]
[latex]n\left(x\right)=4\mathrm{csc}\left(\frac{5\pi }{3}x-\frac{20\pi }{3}\right)[/latex]
Write the equation for the graph in (Figure) in terms of the secant function and give the period and phase shift.
If[latex]\text{ }\mathrm{tan}\text{ }x=3,\text{ }[/latex]find[latex]\text{ }\mathrm{tan}\left(-x\right).[/latex]
If[latex]\text{ }\mathrm{sec}\text{ }x=4,\text{ }[/latex]find[latex]\text{ }\mathrm{sec}\left(-x\right).[/latex]
For the following exercises, graph the functions on the specified window and answer the questions.
Graph[latex]\text{ }m\left(x\right)=\mathrm{sin}\left(2x\right)+\mathrm{cos}\left(3x\right)\text{ }[/latex]on the viewing window[latex]\text{ }\left[-10,10\right]\text{ }[/latex]by[latex]\text{ }\left[-3,3\right].\text{ }[/latex]Approximate the graph’s period.
Graph[latex]\text{ }n\left(x\right)=0.02\mathrm{sin}\left(50\pi x\right)\text{ }[/latex]on the following domains in[latex]\text{ }x:[/latex][latex]\left[0,1\right]\text{ }[/latex]and[latex]\text{ }\left[0,3\right].\text{ }[/latex]Suppose this function models sound waves. Why would these views look so different?
Graph[latex]\text{ }f\left(x\right)=\frac{\mathrm{sin}\text{ }x}{x}\text{ }[/latex]on[latex]\text{ }\left[-0.5,0.5\right]\text{ }[/latex]and explain any observations.
For the following exercises, let[latex]\text{ }f\left(x\right)=\frac{3}{5}\mathrm{cos}\left(6x\right).[/latex]
What is the largest possible value for[latex]\text{ }f\left(x\right)?[/latex]
What is the smallest possible value for[latex]\text{ }f\left(x\right)?[/latex]
Where is the function increasing on the interval[latex]\text{ }\left[0,2\pi \right]?[/latex]
For the following exercises, find and graph one period of the periodic function with the given amplitude, period, and phase shift.
Sine curve with amplitude 3, period[latex]\text{ }\frac{\pi }{3},\text{ }[/latex]and phase shift[latex]\text{ }\left(h,k\right)=\left(\frac{\pi }{4},2\right)[/latex]
Cosine curve with amplitude 2, period[latex]\text{ }\frac{\pi }{6},\text{ }[/latex]and phase shift[latex]\text{ }\left(h,k\right)=\left(-\frac{\pi }{4},3\right)[/latex]
![A graph of one period of a cosine function, graphed over -pi/4 to 0. Range is [1,5], period is pi/6.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3896/2019/03/07134034/CNX_Precalc_Figure_06_03_251.jpg)
For the following exercises, graph the function. Describe the graph and, wherever applicable, any periodic behavior, amplitude, asymptotes, or undefined points.
[latex]f\left(x\right)=5\mathrm{cos}\left(3x\right)+4\mathrm{sin}\left(2x\right)[/latex]
[latex]f\left(x\right)={e}^{\mathrm{sin}t}[/latex]
For the following exercises, find the exact value.
[latex]{\mathrm{sin}}^{-1}\left(\frac{\sqrt{3}}{2}\right)[/latex]
[latex]{\mathrm{tan}}^{-1}\left(\sqrt{3}\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(\mathrm{sin}\left(\pi \right)\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(\mathrm{tan}\left(\frac{7\pi }{4}\right)\right)[/latex]
[latex]\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(1-2x\right)\right)[/latex]
[latex]{\mathrm{cos}}^{-1}\left(-0.4\right)[/latex]
[latex]\mathrm{cos}\left({\mathrm{tan}}^{-1}\left({x}^{2}\right)\right)[/latex]
For the following exercises, suppose[latex]\text{ }\mathrm{sin}\text{ }t=\frac{x}{x+1}.[/latex] Evaluate the following expressions.
[latex]\mathrm{tan}\text{ }t[/latex]
[latex]\mathrm{csc}\text{ }t[/latex]
Given (Figure), find the measure of angle[latex]\text{ }\theta \text{ }[/latex]to three decimal places. Answer in radians.
For the following exercises, determine whether the equation is true or false.
[latex]\mathrm{arcsin}\left(\mathrm{sin}\left(\frac{5\pi }{6}\right)\right)=\frac{5\pi }{6}[/latex]
[latex]\mathrm{arccos}\left(\mathrm{cos}\left(\frac{5\pi }{6}\right)\right)=\frac{5\pi }{6}[/latex]
The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.
From Sections 7 – 9
Chapter Review Exercises
Solving Trigonometric Equations with Identities
For the following exercises, find all solutions exactly that exist on the interval [latex]\left[0,2\pi \right).[/latex]
1. [latex]{\mathrm{csc}}^{2}t=3[/latex]
2. [latex]{\mathrm{cos}}^{2}x=\frac{1}{4}[/latex]
3. [latex]2\text{ }\mathrm{sin}\text{ }\theta =-1[/latex]
4. [latex]\mathrm{tan}\text{ }x\text{ }\mathrm{sin}\text{ }x+\mathrm{sin}\left(-x\right)=0[/latex]
5. [latex]9\text{ }\mathrm{sin}\text{ }\omega -2=4\text{ }{\mathrm{sin}}^{2}\omega[/latex]
6. [latex]1-2\text{ }\mathrm{tan}\left(\omega \right)={\mathrm{tan}}^{2}\left(\omega \right)[/latex]
For the following exercises, use basic identities to simplify the expression.
7. [latex]\mathrm{sec}\text{ }x\text{ }\mathrm{cos}\text{ }x+\mathrm{cos}\text{ }x-\frac{1}{\mathrm{sec}\text{ }x}[/latex]
8. [latex]{\mathrm{sin}}^{3}x+{\mathrm{cos}}^{2}x\text{ }\mathrm{sin}\text{ }x[/latex]
For the following exercises, determine if the given identities are equivalent.
9. [latex]{\mathrm{sin}}^{2}x+{\mathrm{sec}}^{2}x-1=\frac{\left(1-{\mathrm{cos}}^{2}x\right)\left(1+{\mathrm{cos}}^{2}x\right)}{{\mathrm{cos}}^{2}x}[/latex]
10. [latex]{\mathrm{tan}}^{3}x\text{ }{\mathrm{csc}}^{2}x\text{ }{\mathrm{cot}}^{2}x\text{ }\mathrm{cos}\text{ }x\text{ }\mathrm{sin}\text{ }x=1[/latex]
Sum and Difference Identities
For the following exercises, find the exact value.
11. [latex]\mathrm{tan}\left(\frac{7\pi }{12}\right)[/latex]
12. [latex]\mathrm{cos}\left(\frac{25\pi }{12}\right)[/latex]
13. [latex]\mathrm{sin}\left({70}^{\circ }\right)\mathrm{cos}\left({25}^{\circ }\right)-\mathrm{cos}\left({70}^{\circ }\right)\mathrm{sin}\left({25}^{\circ }\right)[/latex]
14. [latex]\mathrm{cos}\left({83}^{\circ }\right)\mathrm{cos}\left({23}^{\circ }\right)+\mathrm{sin}\left({83}^{\circ }\right)\mathrm{sin}\left({23}^{\circ }\right)[/latex]
For the following exercises, prove the identity.
15. [latex]\mathrm{cos}\left(4x\right)-\mathrm{cos}\left(3x\right)\mathrm{cos}x={\mathrm{sin}}^{2}x-4\text{ }{\mathrm{cos}}^{2}x\text{ }{\mathrm{sin}}^{2}x[/latex]
16. [latex]\mathrm{cos}\left(3x\right)-{\mathrm{cos}}^{3}x=-\mathrm{cos}\text{ }x\text{ }{\mathrm{sin}}^{2}x-\mathrm{sin}\text{ }x\text{ }\mathrm{sin}\left(2x\right)[/latex]
For the following exercise, simplify the expression.
17. [latex]\frac{\mathrm{tan}\left(\frac{1}{2}x\right)+\mathrm{tan}\left(\frac{1}{8}x\right)}{1-\mathrm{tan}\left(\frac{1}{8}x\right)\mathrm{tan}\left(\frac{1}{2}x\right)}[/latex]
For the following exercises, find the exact value.
18. [latex]\mathrm{cos}\left({\mathrm{sin}}^{-1}\left(0\right)-{\mathrm{cos}}^{-1}\left(\frac{1}{2}\right)\right)[/latex]
19. [latex]\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(0\right)+{\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)\right)[/latex]
Double-Angle, Half-Angle, and Reduction Formulas
For the following exercises, find the exact value.
20. Find [latex]\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),[/latex] and [latex]\mathrm{tan}\left(2\theta \right)[/latex] given [latex]\mathrm{cos}\text{ }\theta =-\frac{1}{3}[/latex] and [latex]\theta[/latex] is in the interval [latex]\left[\frac{\pi }{2},\pi \right].[/latex]
21. Find [latex]\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),[/latex] and [latex]\mathrm{tan}\left(2\theta \right)[/latex] given [latex]\mathrm{sec}\text{ }\theta =-\frac{5}{3}[/latex] and [latex]\theta[/latex] is in the interval [latex]\left[\frac{\pi }{2},\pi \right].[/latex]
22. [latex]\mathrm{sin}\left(\frac{7\pi }{8}\right)[/latex]
23. [latex]\mathrm{sec}\left(\frac{3\pi }{8}\right)[/latex]
For the following exercises, use (Figure) to find the desired quantities.
24. [latex]\mathrm{sin}\left(2\beta \right),\mathrm{cos}\left(2\beta \right),\mathrm{tan}\left(2\beta \right),\mathrm{sin}\left(2\alpha \right),\mathrm{cos}\left(2\alpha \right),\text{ and }\mathrm{tan}\left(2\alpha \right)[/latex]
25. [latex]\mathrm{sin}\left(\frac{\beta }{2}\right),\mathrm{cos}\left(\frac{\beta }{2}\right),\mathrm{tan}\left(\frac{\beta }{2}\right),\mathrm{sin}\left(\frac{\alpha }{2}\right),\mathrm{cos}\left(\frac{\alpha }{2}\right),\text{ and }\mathrm{tan}\left(\frac{\alpha }{2}\right)[/latex]
For the following exercises, prove the identity.
26. [latex]\frac{2\mathrm{cos}\left(2x\right)}{\mathrm{sin}\left(2x\right)}=\mathrm{cot}\text{ }x-\mathrm{tan}\text{ }x[/latex]
27. [latex]\mathrm{cot}\text{ }x\text{ }\mathrm{cos}\left(2x\right)=-\mathrm{sin}\left(2x\right)+\mathrm{cot}\text{ }x[/latex]
For the following exercises, rewrite the expression with no powers.
28. [latex]{\mathrm{cos}}^{2}x\text{ }{\mathrm{sin}}^{4}\left(2x\right)[/latex]
29. [latex]{\mathrm{tan}}^{2}x\text{ }{\mathrm{sin}}^{3}x[/latex]
Sum-to-Product and Product-to-Sum Formulas
For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.
30. [latex]\mathrm{cos}\left(\frac{\pi }{3}\right)\text{ }\mathrm{sin}\left(\frac{\pi }{4}\right)[/latex]
31. [latex]2\text{ }\mathrm{sin}\left(\frac{2\pi }{3}\right)\text{ }\mathrm{sin}\left(\frac{5\pi }{6}\right)[/latex]
32. [latex]2\text{ }\mathrm{cos}\left(\frac{\pi }{5}\right)\text{ }\mathrm{cos}\left(\frac{\pi }{3}\right)[/latex]
For the following exercises, evaluate the sum by using a product formula. Write the exact answer.
33. [latex]\mathrm{sin}\left(\frac{\pi }{12}\right)-\mathrm{sin}\left(\frac{7\pi }{12}\right)[/latex]
34. [latex]\mathrm{cos}\left(\frac{5\pi }{12}\right)+\mathrm{cos}\left(\frac{7\pi }{12}\right)[/latex]
For the following exercises, change the functions from a product to a sum or a sum to a product.
35. [latex]\mathrm{sin}\left(9x\right)\mathrm{cos}\left(3x\right)[/latex]
36. [latex]\mathrm{cos}\left(7x\right)\mathrm{cos}\left(12x\right)[/latex]
37. [latex]\mathrm{sin}\left(11x\right)+\mathrm{sin}\left(2x\right)[/latex]
38. [latex]\mathrm{cos}\left(6x\right)+\mathrm{cos}\left(5x\right)[/latex]
Solving Trigonometric Equations
For the following exercises, find all exact solutions on the interval [latex]\left[0,2\pi \right).[/latex]
39. [latex]\mathrm{tan}\text{ }x+1=0[/latex]
40. [latex]2\text{ }\mathrm{sin}\left(2x\right)+\sqrt{2}=0[/latex]
For the following exercises, find all exact solutions on the interval [latex]\left[0,2\pi \right).[/latex]
41. [latex]2\text{ }{\mathrm{sin}}^{2}x-\mathrm{sin}\text{ }x=0[/latex]
42. [latex]{\mathrm{cos}}^{2}x-\mathrm{cos}\text{ }x-1=0[/latex]
43. [latex]2\text{ }{\mathrm{sin}}^{2}x+5\text{ }\mathrm{sin}\text{ }x+3=0[/latex]
44. [latex]\mathrm{cos}\text{ }x-5\text{ }\mathrm{sin}\left(2x\right)=0[/latex]
45. [latex]\frac{1}{{\mathrm{sec}}^{2}x}+2+{\mathrm{sin}}^{2}x+4\text{ }{\mathrm{cos}}^{2}x=0[/latex]
For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [latex]\left[0,2\pi \right).[/latex] Round to four decimal places.
46. [latex]\sqrt{3}\text{ }{\mathrm{cot}}^{2}x+\mathrm{cot}\text{ }x=1[/latex]
47. [latex]{\mathrm{csc}}^{2}x-3\text{ }\mathrm{csc}\text{ }x-4=0[/latex]
For the following exercises, graph each side of the equation to find the zeroes on the interval [latex]\left[0,2\pi \right).[/latex]
48. [latex]20\text{ }{\mathrm{cos}}^{2}x+21\text{ }\mathrm{cos}\text{ }x+1=0[/latex]
49. [latex]{\mathrm{sec}}^{2}x-2\text{ }\mathrm{sec}\text{ }x=15[/latex]
Modeling with Trigonometric Equations
For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.
50.
| [latex]x[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] | [latex]3[/latex] | [latex]4[/latex] | [latex]5[/latex] |
| [latex]y[/latex] | [latex]1[/latex] | [latex]6[/latex] | [latex]11[/latex] | [latex]6[/latex] | [latex]1[/latex] | [latex]6[/latex] |
51.
| [latex]x[/latex] | [latex]y[/latex] |
| [latex]0[/latex] | [latex]-2[/latex] |
| [latex]1[/latex] | [latex]1[/latex] |
| [latex]2[/latex] | [latex]-2[/latex] |
| [latex]3[/latex] | [latex]-5[/latex] |
| [latex]4[/latex] | [latex]-2[/latex] |
| [latex]5[/latex] | [latex]1[/latex] |
52.
| [latex]x[/latex] | [latex]y[/latex] |
| [latex]-3[/latex] | [latex]3+2\sqrt{2}[/latex] |
| [latex]-2[/latex] | [latex]3[/latex] |
| [latex]-1[/latex] | [latex]2\sqrt{2}-1[/latex] |
| [latex]0[/latex] | [latex]1[/latex] |
| [latex]1[/latex] | [latex]3-2\sqrt{2}[/latex] |
| [latex]2[/latex] | [latex]-1[/latex] |
| [latex]3[/latex] | [latex]-1-2\sqrt{2}[/latex] |
53. A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?
54. Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?
For the following exercises, construct functions that model the described behavior.
55. A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to [latex]t,[/latex] the month.
56. Daily temperatures in the desert can be very extreme. If the temperature varies from [latex]90\text{°F}[/latex] to [latex]30\text{°F}[/latex] and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.
For the following exercises, find the amplitude, frequency, and period of the given equations.
57. [latex]y=3\text{ }\mathrm{cos}\left(x\pi \right)[/latex]
58. [latex]y=-2\text{ }\mathrm{sin}\left(16x\pi \right)[/latex]
For the following exercises, model the described behavior and find requested values.
59. An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to [latex]t,[/latex] the number of years from now.
60. The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to [latex]t,[/latex] the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.
Practice Test
For the following exercises, simplify the given expression.
1. [latex]\mathrm{cos}\left(-x\right)\mathrm{sin}\text{ }x\text{ }\mathrm{cot}\text{ }x+{\mathrm{sin}}^{2}x[/latex]
2. [latex]\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-2x\right)-\mathrm{sin}\left(-x\right)\mathrm{cos}\left(-2x\right)[/latex]
For the following exercises, find the exact value.
3. [latex]\mathrm{cos}\left(\frac{7\pi }{12}\right)[/latex]
4. [latex]\mathrm{tan}\left(\frac{3\pi }{8}\right)[/latex]
5. [latex]\mathrm{tan}\left({\mathrm{sin}}^{-1}\left(\frac{\sqrt{2}}{2}\right)+{\mathrm{tan}}^{-1}\sqrt{3}\right)[/latex]
6. [latex]2\mathrm{sin}\left(\frac{\pi }{4}\right)\mathrm{sin}\left(\frac{\pi }{6}\right)[/latex]
For the following exercises, find all exact solutions to the equation on [latex]\left[0,2\pi \right).[/latex]
7. [latex]{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1=0[/latex]
8. [latex]{\mathrm{cos}}^{2}x=\mathrm{cos}\text{ }x[/latex]
9. [latex]\mathrm{cos}\left(2x\right)+{\mathrm{sin}}^{2}x=0[/latex]
10. [latex]2\text{ }{\mathrm{sin}}^{2}x-\mathrm{sin}\text{ }x=0[/latex]
11. Rewrite the expression as a product instead of a sum: [latex]\mathrm{cos}\left(2x\right)+\mathrm{cos}\left(-8x\right).[/latex]
12. Find all solutions of [latex]\mathrm{tan}\left(x\right)-\sqrt{3}=0.[/latex]
13. Find the solutions of [latex]{\mathrm{sec}}^{2}x-2\text{ }\mathrm{sec}\text{ }x=15[/latex] on the interval [latex]\left[0,2\pi \right)[/latex] algebraically; then graph both sides of the equation to determine the answer.
14. Find [latex]\mathrm{sin}\left(2\theta \right),\mathrm{cos}\left(2\theta \right),[/latex] and [latex]\mathrm{tan}\left(2\theta \right)[/latex] given [latex]\mathrm{cot}\text{ }\theta =-\frac{3}{4}[/latex] and [latex]\theta[/latex] is on the interval [latex]\left[\frac{\pi }{2},\pi \right].[/latex]
15. Find [latex]\mathrm{sin}\left(\frac{\theta }{2}\right),\mathrm{cos}\left(\frac{\theta }{2}\right),[/latex] and [latex]\mathrm{tan}\left(\frac{\theta }{2}\right)[/latex] given [latex]\mathrm{cos}\text{ }\theta =\frac{7}{25}[/latex] and [latex]\theta[/latex] is in quadrant IV.
16. Rewrite the expression [latex]{\mathrm{sin}}^{4}x[/latex] with no powers greater than 1.
For the following exercises, prove the identity.
17. [latex]{\mathrm{tan}}^{3}x-\mathrm{tan}\text{ }x\text{ }{\mathrm{sec}}^{2}x=\mathrm{tan}\left(-x\right)[/latex]
18. [latex]\mathrm{sin}\left(3x\right)-\mathrm{cos}\text{ }x\text{ }\mathrm{sin}\left(2x\right)={\mathrm{cos}}^{2}x\text{ }\mathrm{sin}\text{ }x-{\mathrm{sin}}^{3}x[/latex]
19. [latex]\frac{\mathrm{sin}\left(2x\right)}{\mathrm{sin}\text{ }x}-\frac{\mathrm{cos}\left(2x\right)}{\mathrm{cos}\text{ }x}=\mathrm{sec}\text{ }x[/latex]
20. Plot the points and find a function of the form [latex]y=A\mathrm{cos}\left(Bx+C\right)+D[/latex] that fits the given data.
| [latex]x[/latex] | [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] | [latex]3[/latex] | [latex]4[/latex] | [latex]5[/latex] |
| [latex]y[/latex] | [latex]-2[/latex] | [latex]2[/latex] | [latex]-2[/latex] | [latex]2[/latex] | [latex]-2[/latex] | [latex]2[/latex] |
21. The displacement [latex]h\left(t\right)[/latex] in centimeters of a mass suspended by a spring is modeled by the function [latex]h\left(t\right)=\frac{1}{4}\text{ }\mathrm{sin}\left(120\pi t\right),[/latex] where [latex]t[/latex] is measured in seconds. Find the amplitude, period, and frequency of this displacement.
22. A woman is standing 300 feet away from a 2000-foot building. If she looks to the top of the building, at what angle above horizontal is she looking? A bored worker looks down at her from the 15th floor (1500 feet above her). At what angle is he looking down at her? Round to the nearest tenth of a degree.
23. Two frequencies of sound are played on an instrument governed by the equation [latex]n\left(t\right)=8\text{ }\mathrm{cos}\left(20\pi t\right)\mathrm{cos}\left(1000\pi t\right).[/latex] What are the period and frequency of the “fast” and “slow” oscillations? What is the amplitude?
24. The average monthly snowfall in a small village in the Himalayas is 6 inches, with the low of 1 inch occurring in July. Construct a function that models this behavior. During what period is there more than 10 inches of snowfall?
25. A spring attached to a ceiling is pulled down 20 cm. After 3 seconds, wherein it completes 6 full periods, the amplitude is only 15 cm. Find the function modeling the position of the spring [latex]t[/latex] seconds after being released. At what time will the spring come to rest? In this case, use 1 cm amplitude as rest.
26. Water levels near a glacier currently average 9 feet, varying seasonally by 2 inches above and below the average and reaching their highest point in January. Due to global warming, the glacier has begun melting faster than normal. Every year, the water levels rise by a steady 3 inches. Find a function modeling the depth of the water [latex]t[/latex] months from now. If the docks are 2 feet above current water levels, at what point will the water first rise above the docks?
27. [latex]D\left(t\right)=2\text{ }\mathrm{cos}\left(\frac{\pi }{6}t\right)+108+\frac{1}{4}t,[/latex] 93.5855 months (or 7.8 years) from now
