Graphs of the Sine and Cosine Function
1. The sine and cosine functions have the property that [latex]f(x+P)=f(x)[/latex] for a certain P. This means that the function values repeat for every P units on the x-axis.
2. The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically.
3. amplitude: [latex]\frac{2}{3}[/latex]; period: 2π; midline: [latex]y=0[/latex]; maximum: [latex]y=23[/latex] occurs at [latex]x=0[/latex]; minimum: [latex]y=−23[/latex] occurs at [latex]x=\pi[/latex]; for one period, the graph starts at 0 and ends at 2π
![A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004034/CNX_Precalc_Figure_06_01_202.jpg)
4. amplitude: 4; period: 2π; midline: [latex]y=0[/latex]; maximum [latex]y=4[/latex] occurs at [latex]x=\frac{\pi}{2}[/latex]; minimum: [latex]y=−4[/latex] occurs at [latex]x=\frac{3\pi}{2}[/latex]; one full period occurs from [latex]x=0[/latex] to [latex]x=2π[/latex]
![A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004035/CNX_Precalc_Figure_06_01_204.jpg)
5. amplitude: 1; period: π; midline: y=0; maximum: y=1 occurs at [latex]x=\pi[/latex]; minimum: [latex]y=−1[/latex] occurs at [latex]x=\frac{\pi}{2}[/latex]; one full period is graphed from [latex]x=0[/latex] to [latex]x=\pi[/latex]
![A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004037/CNX_Precalc_Figure_06_01_206.jpg)
6. amplitude: 4; period: 2; midline: [latex]y=0[/latex]; maximum: [latex]y=4[/latex] occurs at [latex]x=0[/latex]; minimum: [latex]y=−4[/latex] occurs at [latex]x=1[/latex]
![A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004039/CNX_Precalc_Figure_06_01_208.jpg)
7. amplitude: 3; period: [latex]\frac{\pi}{4}[/latex]; midline: [latex]y=5[/latex]; maximum: [latex]y=8[/latex] occurs at [latex]x=0.12[/latex]; minimum: [latex]y=2[/latex] occurs at [latex]x=0.516[/latex]; horizontal shift: −4; vertical translation 5; one period occurs from [latex]x=0[/latex] to [latex]x=\frac{\pi}{4}[/latex]
![A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004041/CNX_Precalc_Figure_06_01_210.jpg)
8. amplitude: 5; period: [latex]\frac{2\pi}{5}; midline: [latex]y=−2[/latex]; maximum: [latex]y=3[/latex] occurs at [latex]x=0.08[/latex]; minimum: [latex]y=−7[/latex] occurs at [latex]x=0.71[/latex]; phase shift:−4; vertical translation:−2; one full period can be graphed on [latex]x=0[/latex] to [latex]x=\frac{2\pi}{5}[/latex]
![A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004042/CNX_Precalc_Figure_06_01_212.jpg)
9. amplitude: 1; period: 2π; midline: y=1; maximum:[latex]y=2[/latex] occurs at [latex]x=2.09[/latex]; maximum:[latex]y=2[/latex] occurs at[latex]t=2.09[/latex]; minimum:[latex]y=0[/latex] occurs at [latex]t=5.24[/latex]; phase shift: [latex]−\frac{\pi}{3}[/latex]; vertical translation: 1; one full period is from [latex]t=0[/latex] to [latex]t=2π[/latex]
![A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004044/CNX_Precalc_Figure_06_01_214.jpg)
10. amplitude: 1; period: 4π; midline: [latex]y=0[/latex]; maximum: [latex]y=1[/latex] occurs at [latex]t=11.52[/latex]; minimum: [latex]y=−1[/latex] occurs at [latex]t=5.24[/latex]; phase shift: −[latex]\frac{10\pi}{3}[/latex]; vertical shift: 0
![A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27004047/CNX_Precalc_Figure_06_01_216.jpg)
11. amplitude: 2; midline: [latex]y=−3[/latex]; period: 4; equation: [latex]f(x)=2\sin\left(\frac{\pi}{2}x\right)−3[/latex]
12. amplitude: 2; period: 5; midline: [latex]y=3[/latex]; equation: [latex]f(x)=−2\cos\left(\frac{2\pi}{5}x\right)+3[/latex]
13. amplitude: 4; period: 2; midline: [latex]y=0[/latex]; equation: [latex]f(x)=−4\cos\left(\pi\left(x−\frac{\pi}{2}\right)\right)[/latex]
14. amplitude: 2; period: 2; midline [latex]y=1[/latex]; equation: [latex]f(x)=2\cos\left(\frac{\pi}{x}\right)+1[/latex]
15. [latex]\frac{\pi}{6},\frac{5\pi}{6}[/latex]
16. [latex]\frac{\pi}{4},\frac{3\pi}{4}[/latex]
17. [latex]\frac{3\pi}{2}[/latex]
18. [latex]\frac{\pi}{2},\frac{3\pi}{2}[/latex]
19. [latex]\frac{\pi}{2},\frac{3\pi}{2}[/latex]
20. [latex]\frac{\pi}{6},\frac{11\pi}{6}[/latex]
21.
a. Amplitude: 12.5; period: 10; midline: [latex]y=13.5[/latex];
b. [latex]h(t)=12.5\sin\left(\frac{\pi}{5}\left(t−2.5\right)\right)+13.5;[/latex]
c. 26 ft
Graphs of Other Trigonometric Functions
1. Since [latex]y=\csc x[/latex] is the reciprocal function of [latex]y=\sin x[/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\sin x[/latex] to obtain the y-coordinates of [latex]y=\csc x[/latex]. The x-intercepts of the graph [latex]y=\sin x[/latex] are the vertical asymptotes for the graph of [latex]y=\csc x[/latex].
2. Answers will vary. Using the unit circle, one can show that [latex]\tan(x+\pi)=\tan x[/latex].
3. The period is the same: 2π.
4. I
5. IV
6. II
7. III
8. period: 8; horizontal shift: 1 unit to left
9. 1.5
10. 5
11. [latex]−\cot x\cos x−\sin x[/latex]
12. stretching factor: 2; period: [latex]\frac{\pi}{4}[/latex]; asymptotes: [latex]x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8[/latex], where k is an integer
13. stretching factor: 6; period: 6; asymptotes: [latex]x=3k[/latex], where k is an integer
14. stretching factor: 1; period: π; asymptotes: [latex]x=πk[/latex], where k is an integer
15. Stretching factor: 1; period: π; asymptotes: [latex]x=\frac{\pi}{4}+{\pi}k[/latex], where k is an integer
16. stretching factor: 2; period: 2π; asymptotes: [latex]x=πk[/latex], where k is an integer
17. stretching factor: 4; period: [latex]\frac{2\pi}{3}[/latex]; asymptotes: [latex]x=\frac{\pi}{6}k[/latex], where k is an odd integer
18. stretching factor: 7; period: [latex]\frac{2\pi}{5}[/latex]; asymptotes: [latex]x=\frac{\pi}{10}k[/latex], where k is an odd integer
19. stretching factor: 2; period: 2π; asymptotes: [latex]x=−\frac{\pi}{4}+\pi k[/latex], where k is an integer
20. stretching factor: [latex]\frac{7}{5}[/latex]; period: 2π; asymptotes: [latex]x=\frac{\pi}{4}+\pi[/latex]k, where k is an integer
21. [latex]y=\tan\left(3\left(x−\frac{\pi}{4}\right)\right)+2[/latex]
22. [latex]f(x)=\csc(2x)[/latex]
23. [latex]f(x)=\csc(4x)[/latex]
24. [latex]f(x)=2\csc x[/latex]
25. [latex]f(x)=\frac{1}{2}\tan(100\pi x)[/latex]
26. [latex]f(x)=\frac{\csc(x)}{\sec(x)}[/latex]
27. a. [latex](−\frac{\pi}{2}\text{,}\frac{\pi}{2})[/latex];
b.
c. [latex]x=−\frac{\pi}{2}[/latex] and [latex]x=\frac{\pi}{2}[/latex]; the distance grows without bound as |x| approaches [latex]\frac{\pi}{2}[/latex]—i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
d. 3; when [latex]x=−\frac{\pi}{3}[/latex], the boat is 3 km away;
e. 1.73; when [latex]x=\frac{\pi}{6}[/latex], the boat is about 1.73 km away;
f. 1.5 km; when [latex]x=0[/latex].
Note: the final answer (labeled 57 in the original, covering h(x)=2tan(πx/120)) has no corresponding renumbered problem, since the exercise list only went up to item 27 in this section.