Trigonometry

1. Draw the graph of the function $f(x)=\sin x$ and determine the following properties:
   

   (a)   find the minimum value and the maximum value,

   (b)   find the period.

   Solutions:    (a)  min. value: $-1$, max. value: $1$;     (b)  period: $2\pi$
2. Draw the graph of the function $f(x)=2\cos x+1$ and determine the following properties:
   

   (a)   find the minimum value and the maximum value,

   (b)   find the period.

   Solutions:    (a)  min. value: $-1$, max. value: $3$;     (b)  period: $2\pi$
3. Draw the graph of the function $f(x)=\sin 2x+3$ and determine the following properties:
   

   (a)   find the domain and range,

   (b)   find the period,

   (c)   write down minima and maxima.

   Solutions:    (a)  domain: $\mathbb{R}$, range: $[2,4]$;     (b)  period: $\pi$;     (c)  min.: $(-\frac{\pi }{4}+k\pi ,2)$, max.:$(\frac{\pi }{4}+k\pi ,4),k\in \mathbb{Z}$
4. Draw the graph of the function $f(x)=\sin 3x$ and determine the following properties:
   

   (a)   find $x$-intercepts,

   (b)   find the period,

   (c)   write down minima and maxima.

   Solutions:    (a)  $x$-intercepts: $\frac{k\pi }{3},k\in \mathbb{Z}$;     (b)  period: $\frac{2\pi }{3}$;     (c)  min.: $(-\frac{\pi }{6}+\frac{2k\pi }{3},-1)$, max.:$(\frac{\pi }{6}+\frac{2k\pi }{3},1),k\in \mathbb{Z}$
5. Draw the graph of the function $f(x)=\cos 3x$ and determine the following properties:
   

   (a)   find all $x$-intercepts on $[0,2\pi ]$,

   (b)   write down minima and maxima on $[0,2\pi ]$.

   Solutions:    (a)  $x$-intercepts: $\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6},\frac{7\pi }{6},\frac{3\pi }{2},\frac{11\pi }{6}$;     (b)  min.: $(\frac{\pi }{3},-1),(\pi ,-1),(\frac{5\pi }{3},-1)$, max.:$(0,1),(\frac{2\pi }{3},1),(\frac{4\pi }{3},1),(2\pi ,1)$
6. Draw the graph of the function $f(x)=4\cos (x-\frac{\pi }{6})$ and determine the following properties:
   

   (a)   find $y$-intercept,

   (b)   find all $x$-intercepts on $[0,2\pi ]$,

   (c)   write down the greatest and the least value, and state the smallest non-negative value of $x$ for which they occur.

   Solutions:    (a)  $y$-intercept: $2\sqrt{3}$;     (b)  $x$-intercepts: $\frac{2\pi }{3},\frac{5\pi }{3}$;     (c)  min. value $-4$ occurs at $x=\frac{7\pi }{6}$, max. value $4$ occurs at $x=\frac{\pi }{6}$
7. Draw the graph of the function $f(x)=\tan x$ and determine the following properties:
   

   (a)   find $x$-intercepts,

   (b)   find the vertical asymptotes.

   Solutions:    (a)  $x$-intercepts: $x=k\pi ,k\in \mathbb{Z}$;     (b)  vertical asymptotes: $x=\frac{\pi }{2}+k\pi ,k\in \mathbb{Z}$
8. Draw the graph of the function $f(x)=\tan \frac{x}{2}$ and determine the following properties:
   

   (a)   find $x$-intercepts,

   (b)   find the vertical asymptotes.

   Solutions:    (a)  $x$-intercepts: $x=2k\pi ,k\in \mathbb{Z}$;     (b)  vertical asymptotes: $x=\pi +2k\pi ,k\in \mathbb{Z}$
9. Draw the graphs of the following functions:
   

   (a)   $y=|\sin x|$

   (b)   $y=|2\cos x+1|$

10. Draw the graphs of the following functions:
    

    (a)   $y=\sin |x|$

    (b)   $y=\tan |x|$

11. Use your GDC to draw the graph of the function $y={\log }_2(\sin x+1)$ and then:
    

    (a)   find $x$-intercepts,

    (b)   find the vertical asymptotes,

    (c)   find minima and maxima.

    Solutions:    (a)  $x$-intercepts: $x=k\pi ,k\in \mathbb{Z}$;     (b)  vertical asymptotes: $x=\frac{3\pi }{2}+2k\pi ,k\in \mathbb{Z}$;     (c)  maxima: $(\frac{\pi }{2}+2k\pi ,1),k\in \mathbb{Z}$, minima don't exist
12. Use your GDC to draw the graph of the function $f(x)=\sin x+\sqrt{3}\cos x$. This function can be written in the form $f(x)=A\sin (x+B)$. Find the values of the constants $A$ and $B$.
    Solutions:    $A=2,B=\frac{\pi }{3}$;     $f(x)=2\sin (x+\frac{\pi }{3})$
13. Use your GDC to draw the graph of the function $f(x)={\sin }^{2}x$. Then write this function in the form $f(x)=A\cos Bx+C$. (Hint: Use the GDC to test your solution.)
    Solutions:    $f(x)=-\frac{1}{2}\cos 2x+\frac{1}{2}$
14. Calculate the values of the following trigonometric functions. Your results should be exact.
    

    (a)   $\sin {120}^{\circ }$

    (b)   $\cos {135}^{\circ }$

    (c)   $\tan {150}^{\circ }$

    Solutions:    (a)  $\cdots =\frac{\sqrt{3}}{2}$;     (b)  $\cdots =-\frac{\sqrt{2}}{2}$;     (c)  $\cdots =-\frac{\sqrt{3}}{3}$
15. Calculate the exact values of the following trigonometric functions:
    

    (a)   $\cos {210}^{\circ }$

    (b)   $\tan {225}^{\circ }$

    (c)   $\sin {330}^{\circ }$

    Solutions:    (a)  $\cdots =-\frac{\sqrt{3}}{2}$;     (b)  $\cdots =1$;     (c)  $\cdots =-\frac{1}{2}$
16. Calculate the exact values of the following trigonometric functions:
    

    (a)   $\sin {405}^{\circ }$

    (b)   $\tan {480}^{\circ }$

    (c)   $\cos {540}^{\circ }$

    (d)   $\sin {1290}^{\circ }$

    Solutions:    (a)  $\cdots =\frac{\sqrt{2}}{2}$;     (b)  $\cdots =-\sqrt{3}$;     (c)  $\cdots =-1$;     (d)  $\cdots =-\frac{1}{2}$
17. Calculate the exact values of the following trigonometric functions:
    

    (a)   $\sin (-{60}^{\circ })$

    (b)   $\tan (-{135}^{\circ })$

    (c)   $\cos (-{270}^{\circ })$

    Solutions:    (a)  $\cdots =-\frac{\sqrt{3}}{2}$;     (b)  $\cdots =1$;     (c)  $\cdots =0$
18. Calculate the exact values of the following trigonometric functions:
    

    (a)   $\sin \frac{7\pi }{6}$

    (b)   $\tan \frac{11\pi }{3}$

    (c)   $\cos (-7\pi )$

    Solutions:    (a)  $\cdots =-\frac{1}{2}$;     (b)  $\cdots =-\sqrt{3}$;     (c)  $\cdots =-1$
19. Express each of the following as trigonometric function of an acute angle:
    

    (a)   $\cos {190}^{\circ }$

    (b)   $\sin {500}^{\circ }$

    (c)   $\tan \frac{7\pi }{5}$

    Solutions:    (a)  $\cdots =-\cos {10}^{\circ }$;     (b)  $\cdots =\sin {40}^{\circ }$;     (c)  $\cdots =\tan \frac{2\pi }{5}=\tan {72}^{\circ }$
20. Find $\sin \alpha$ given that $\alpha$ is an acute angle and $\cos \alpha =\frac{3}{5}$.
    (Note: Use the standard trigonometric identities to find the exact result.)
    Solutions:    $\sin \alpha =\frac{4}{5}$
21. Given that $\sin x=\frac{15}{17}$ and ${90}^{\circ }<x<{180}^{\circ }$, find the exact values of:
    

    (a)   $\cos x$

    (b)   $\tan x$

    Solutions:    (a)  $\cos x=-\frac{8}{17}$;     (b)  $\tan x=-\frac{15}{8}$
22. Given that $\cos x=-\frac{2}{3}$ and $\pi <x<2\pi$, find the exact values of:
    

    (a)   $\sin x$

    (b)   $\tan x$

    Solutions:    (a)  $\sin x=-\frac{\sqrt{5}}{3}$;     (b)  $\tan x=\frac{\sqrt{5}}{2}$
23. Given that $\tan x=-2\sqrt{2}$ and $0<x<\pi$, find the exact values of:
    

    (a)   $\cos x$

    (b)   $\sin x$

    Solutions:    (a)  $\cos x=-\frac{1}{3}$;     (b)  $\sin x=\frac{2\sqrt{2}}{3}$
24. Simplify the following expressions using the standard trigonometric identities:
    

    (a)   ${\displaystyle \frac{1-{\cos }^{2}x}{\sin x}}$

    (b)   ${\displaystyle (\frac{1}{\sin x}-\sin x)\cdot \frac{1}{{\cos }^{2}x}}$

    (c)   ${\displaystyle (\sin x-\frac{1}{\sin x})\cdot \tan x}$

    Solutions:    (a)  $\cdots =\sin x$;     (b)  $\cdots =\frac{1}{\sin x}$;     (c)  $\cdots =-\cos x$
25. Simplify the following expressions:
    

    (a)   ${\displaystyle \frac{{\tan }^{2}x}{{\sin }^{2}x}\cdot \frac{1-{\sin }^{2}x}{1+{\tan }^{2}x}}$

    (b)   ${\displaystyle \frac{{\displaystyle \frac{1}{\cos x}}-\cos x}{\tan x}+\frac{1}{\sin x({\tan }^{2}x+1)}}$

    (c)   ${\displaystyle (\tan x+\frac{1}{\tan x}-\frac{1}{\sin x})(1+\cos x)}$

    Solutions:    (a)  $\cdots ={\cos }^{2}x$;     (b)  $\cdots =\frac{1}{\sin x}$;     (c)  $\cdots =\tan x$
26. Simplify the following expressions using the standard trigonometric identities and the double angles formulae:
    

    (a)   ${\displaystyle \sin 2x\cdot \frac{{\displaystyle \frac{1}{\sin x}}-\sin x}{\cos x}}$

    (b)   ${\displaystyle \frac{2\tan x-\sin 2x}{1-\cos 2x}}$

    (c)   ${\displaystyle \frac{{\displaystyle \frac{1}{\cos 2x}}-{\cos }^{2}x+{\sin }^{2}x}{\tan 2x}}$

    (d)   ${\displaystyle \frac{\tan x+\sin x}{\tan x\sin 2x}\cdot (1-\cos x)}$

    Solutions:    (a)  $\cdots =2{\cos }^{2}x$;     (b)  $\cdots =\tan x$;     (c)  $\cdots =\sin 2x$;     (d)  $\cdots =\frac{1}{2}\tan x$
27. Solve the following equations for $-{180}^{\circ }⩽x<{180}^{\circ }$. Write the solutions in degrees.
    

    (a)   $\sin x=\frac{1}{2}$

    (b)   $\sin x=-\frac{\sqrt{2}}{2}$

    (c)   $\cos x=\frac{1}{2}$

    (d)   $\cos x=0$

    Solutions:    (a)  $x_1={30}^{\circ },x_2={150}^{\circ }$;     (b)  $x_1=-{45}^{\circ },x_2=-{135}^{\circ }$;     (c)  $x_1={60}^{\circ },x_2=-{60}^{\circ }$;     (d)  $x_1={90}^{\circ },x_2=-{90}^{\circ }$
28. Solve the following equations for $0⩽x<2\pi$. Write the solutions in radians.
    

    (a)   $\sin x=\frac{1}{2}$

    (b)   $\sin x=\frac{\sqrt{2}}{2}$

    (c)   $\cos x=\frac{\sqrt{2}}{2}$

    (d)   $\cos x=-\frac{1}{2}$

    Solutions:    (a)  $x_1=\frac{\pi }{6},x_2=\frac{5\pi }{6}$;     (b)  $x_1=\frac{\pi }{4},x_2=\frac{3\pi }{4}$;     (c)  $x_1=\frac{\pi }{4},x_2=\frac{7\pi }{4}$;     (d)  $x_1=\frac{5\pi }{6},x_2=\frac{7\pi }{6}$
29. Find all possible solutions of the following equations. Write the solutions in radians.
    

    (a)   $\sin x=\frac{1}{2}$

    (b)   $\sin x=-1$

    (c)   $\cos x=\frac{\sqrt{3}}{2}$

    (d)   $\cos x=-\frac{1}{2}$

    Solutions:    (a)  $x_1=\frac{\pi }{6}+2k\pi ,x_2=\frac{5\pi }{6}+2k\pi ,(k\in \mathbb{Z})$;     (b)  $x_1=-\frac{\pi }{2}+2k\pi ,(k\in \mathbb{Z})$;     (c)  $x_1=\pm \frac{\pi }{6}+2k\pi ,(k\in \mathbb{Z})$;     (d)  $x_1=\pm \frac{2\pi }{3}+2k\pi ,(k\in \mathbb{Z})$
30. Find all possible solutions of the following equations. Write the solutions in radians.
    

    (a)   $\sin 2x=\frac{1}{2}$

    (b)   $\sin (x+\frac{\pi }{4})=\frac{\sqrt{2}}{2}$

    (c)   $\cos 5x=-\frac{\sqrt{3}}{2}$

    (d)   $\cos (3x-\frac{\pi }{6})=\sqrt{2}$

    Solutions:    (a)  $x_1=\frac{\pi }{12}+k\pi ,x_2=\frac{5\pi }{12}+k\pi ,(k\in \mathbb{Z})$;     (b)  $x_1=2k\pi ,x_2=\frac{\pi }{2}+2k\pi ,(k\in \mathbb{Z})$;     (c)  $x_1=\pm \frac{\pi }{6}+\frac{2}{5}k\pi ,(k\in \mathbb{Z})$;     (d)  this equation has no solutions (cosine can't be greater than 1)
31. Find all possible solutions of the following equations. Write the solutions in degrees.
    

    (a)   $\sin 2x=\frac{1}{2}$

    (b)   $\sin (5x+{40}^{\circ })=\frac{\sqrt{3}}{2}$

    (c)   $\cos 6x=-1$

    (d)   $\cos (x+{25}^{\circ })=\frac{1}{3}$

    Solutions:    (a)  $x_1={15}^{\circ }+{180}^{\circ }k,x_2={75}^{\circ }+{180}^{\circ }k,(k\in \mathbb{Z})$;     (b)  $x_1=4^{\circ }+{72}^{\circ }k,x_2={16}^{\circ }+{72}^{\circ }k,(k\in \mathbb{Z})$;     (c)  $x_1={30}^{\circ }+{60}^{\circ }k,(k\in \mathbb{Z})$;     (d)  $x_1\approx {45}^{\circ }{32}^{\prime }+{360}^{\circ }k,x_2\approx -{95}^{\circ }{32}^{\prime }+{360}^{\circ }k,(k\in \mathbb{Z})$
32. Find all solutions for $x\in [0,{360}^{\circ })$. Write the solutions in degrees and minutes.
    

    (a)   $\sin 2x=\frac{1}{2}$

    (b)   $\sin (3x-{15}^{\circ }{30}^{\prime })=0$

    (c)   $\sin \frac{3x}{2}=\frac{2}{3}$

    Solutions:    (a)  $x_1={15}^{\circ },x_2={75}^{\circ },x_3={195}^{\circ },x_4={255}^{\circ }$;     (b)  $x_1=5^{\circ }{10}^{\prime },x_2={65}^{\circ }{10}^{\prime },x_3={125}^{\circ }{10}^{\prime },x_4={185}^{\circ }{10}^{\prime },x_5={245}^{\circ }{10}^{\prime },x_6={305}^{\circ }{10}^{\prime }$;     (c)  $x_1\approx {27}^{\circ }{52}^{\prime },x_2\approx {92}^{\circ }{8}^{\prime },x_3\approx {267}^{\circ }{52}^{\prime },x_4\approx {332}^{\circ }{8}^{\prime }$
33. Find all possible solutions of the following equations. Write the solutions in radians.
    

    (a)   $\tan x=1$

    (b)   $\tan x=\sqrt{3}$

    (c)   $\tan 2x=-1$

    (d)   $\tan 2x=3$

    Solutions:    (a)  $x_1=\frac{\pi }{4}+k\pi ,(k\in \mathbb{Z})$;     (b)  $x_1=\frac{\pi }{3}+k\pi ,(k\in \mathbb{Z})$;     (c)  $x_1=-\frac{\pi }{8}+\frac{1}{2}k\pi ,(k\in \mathbb{Z})$;     (d)  $x_1\approx 0.6245+\frac{k\pi }{2},(k\in \mathbb{Z})$
34. Find all possible solutions of the following equations. Write the solutions in degrees.
    

    (a)   $\tan (x+{25}^{\circ })=\frac{\sqrt{3}}{3}$

    (b)   $\tan 9x=0$

    (c)   $\tan (2x-{72}^{\circ })=-\frac{1}{2}$

    Solutions:    (a)  $x_1=5^{\circ }+{180}^{\circ }k,(k\in \mathbb{Z})$;     (b)  $x_1={20}^{\circ }k,(k\in \mathbb{Z})$;     (c)  $x_1\approx {22}^{\circ }{43}^{\prime }+{90}^{\circ }k,(k\in \mathbb{Z})$
35. Find all solutions for $x\in (-{180}^{\circ },{180}^{\circ }]$. Write the solutions in degrees and minutes.
    

    (a)   $\tan (x-{15}^{\circ })=\sqrt{3}$

    (b)   $\tan \frac{3x}{2}=-3$

    Solutions:    (a)  $x_1={75}^{\circ },x_2=-{105}^{\circ }$;     (b)  $x_1\approx {72}^{\circ }{17}^{\prime },x_2\approx -{47}^{\circ }{43}^{\prime },x_3\approx -{167}^{\circ }{43}^{\prime }$
36. Find all solutions of the following equations:
    

    (a)   ${\cos }^{2}x-4\cos x+3=0$

    (b)   $2{\sin }^{2}x-11\sin x=6$

    (c)   $2{\sin }^{2}x+3\cos x=3$

    Solutions:    (a)  $x=2k\pi (k\in \mathbb{Z})$;     (b)  $x_1=-\frac{\pi }{6}+2k\pi ,x_2=-\frac{5\pi }{6}+2k\pi (k\in \mathbb{Z})$;     (c)  $x_1=2k\pi ,x_2=\pm \frac{\pi }{3}+2k\pi (k\in \mathbb{Z})$
37. Find all solutions of the following equations:
    

    (a)   ${\sin }^{2}x+2\sin x=0$

    (b)   ${\tan }^{2}x-\tan x=0$

    (c)   $2\sin x\cos x-2\sin x+\sqrt{3}\cos x-\sqrt{3}=0$

    Solutions:    (a)  $x=k\pi (k\in \mathbb{Z})$;     (b)  $x_1=k\pi ,x_2=\frac{\pi }{4}+k\pi (k\in \mathbb{Z})$;     (c)  $x_1=2k\pi ,x_2=-\frac{\pi }{3}+2k\pi ,x_3=-\frac{2\pi }{3}+2k\pi (k\in \mathbb{Z})$
38. Find all solutions of the following equations:
    

    (a)   $4{\sin }^{2}x\cos x=8\cos x-7\sin 2x$

    (b)   $\tan x{\cos }^{2}x-\frac{5}{2}\tan x=-\frac{9}{2}\tan x\cos x$

    Solutions:    (a)  $x_1=\frac{\pi }{2}+k\pi ,x_2=\frac{\pi }{6}+2k\pi ,x_2=\frac{5\pi }{6}+2k\pi (k\in \mathbb{Z})$;     (b)  $x_1=k\pi ,x_2=\pm \frac{\pi }{3}+2k\pi (k\in \mathbb{Z})$
39. Find all solutions of the following equations:
    

    (a)   $\sin x+\sqrt{3}\cos x=0$

    (b)   ${\sin }^{2}x+\sin 2x-3{\cos }^{2}x=0$

    (c)   $6{\sin }^{2}x-\sin 2x-2{\cos }^{2}x=1$

    Solutions:    (a)  $x_1=-\frac{\pi }{3}+k\pi (k\in \mathbb{Z})$;     (b)  $x_1=\frac{\pi }{4}+k\pi ,x_2=-\arctan 3+k\pi (k\in \mathbb{Z})$;     (c)  $x_1=\frac{\pi }{4}+k\pi ,x_2=-\arctan \frac{3}{5}+k\pi (k\in \mathbb{Z})$

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