Free Printable Trig Identities and Equations

1.

Find tan u.

a)

$\frac{\sqrt{5}}{2}$

b)

$\sqrt{5}$

c)

$\sqrt{3}$

d)

$\frac{\sqrt{5}}{3}$

2.

a)

$\frac{-4\sqrt{15}}{15}$

b)

$\frac{4\sqrt{15}}{15}$

c)

$\frac{15}{4}$

d)

$\frac{-15}{4}$

3.

Find sec(x).

a)

$\frac{-\sqrt{65}}{7}$

b)

$\frac{-\sqrt{65}}{4}$

c)

$\frac{-4}{\sqrt{65}}$

d)

$\frac{-7}{\sqrt{65}}$

4.

Simplify

a)

sin²θ

b)

cosθ

c)

tanθ

d)

1- sin²θ

5.

Simplify

a)

-1

b)

sin θ

c)

csc θ

d)

1

6.

Simplify (Hint: you will need to FOIL first)

a)

sinθ

b)

cot²θ

c)

tan²θ

d)

cos²θ

7.

Simplify: tanxcotx-cos²x

a)

tanx

b)

cotx

c)

sin²x

d)

cos²x

8.

Simplify

a)

csc²θ

b)

sec²θ

c)

cscθ

d)

1

9.

Simplify: sin x cos²x – sin x

a)

-sin³x

b)

-sin²x

c)

-cos³x

d)

-cos²x

e)

tan x

10.

Use the fundamental identities to simplify the expression.

a)

$\sin\left(\theta\right)$

b)

$\cos\left(\theta\right)$

c)

$\tan\left(\theta\right)$

d)

$\csc\left(\theta\right)$

e)

$\cot\left(\theta\right)$

11.

What would be the best first step to prove the identity?

a)

work with the left side; get a common denominator

b)

work with the left side; multiply the fractions

c)

work with the right side; use the quotient identities to change tan(x) to sin(x)/cos(x) and cot(x) to cos(x)/sin(x)

d)

work with the right side; use the Pythagorean identities

12.

What would be the best first step to verifying the trig identity?

a)

work with the left side; use Pythagorean identity to change tan²x

b)

work with left side; use reciprocal identity to change cos x to 1/sin x

c)

work with the right side; change sec x to 1/cos x using reciprocal identity

d)

work with the right side; multiply sec x by sin²x + cos²x

e)

square both sides to get a Pythagorean identity

13.

What would be the best first step to proving this identity?

a)

factor out cot x from left side

b)

use a Pythagorean identity to change sec²x

c)

multiply both sides by the conjugate of the left side

d)

divide both sides by sec²x

e)

change cot x to cos x / sin x using reciprocal identity

14.

Of the following, which would be the best first step to proving this identity?

a)

change cot(pi/2 - x) to tan x using co-function identity

b)

change sec x to 1/cos x using reciprocal identity

c)

square both sides to try to get Pythagorean identity

d)

change cot(pi/2 - x) to 1/tan(pi/2 - x) using a reciprocal identity

e)

change cot (pi/2 - x) to -cot(x - pi/2) using even-odd identity

15.

What would be the best first step to verifying this identity?

a)

working with left side, use Pythagorean identity to sin²x to (1 - cos²x)

b)

working with left side, use reciprocal identity to change cos³x to 1/sec³x

c)

working with left side, factor out cosx and then replace cos²x to (1 - sin²x) and sin²x to (1 - cos²x)

d)

working with right side, distribute the cos x into the parenthetical expression and then use reciprocal identity to change cos x to 1/sec x

e)

working with right side, factor out sin²x from parenthetical expression and then use Pythagorean identity to change (1 - sin²x) to cos²x

16.

Solve
cosθ = - √(3)/2
on θ∈[0, 2π)

a)

θ = π /3, 2π /3

b)

θ = 2π /3, 4π /3

c)

θ = π /6, 5π /6

d)

θ = 5π /6, 7π /6

17.

Solve on the domain [0,2π)
cos x + 1 = 0

a)

0

b)

No solution

c)

π

d)

3π/2

18.

a)

A

b)

B

c)

C

d)

D

19.

Solve on the Interval [0,2π)
tan(x)+1=2

a)

0 and π

b)

3π/4 and 7π/4

c)

π/4 and 5π/4

d)

3π/4 and 5π/4

20.

Solve for x

don't forget cot = cos/sin

a)

π

b)

π/3

c)

π/2

d)

-π

21.

a)

A

b)

B

c)

C

d)

D

22.

Solve on the domain [0, 2π)
4sin²x = 3

a)

π/6, 11π/6

b)

π/3, 2π/3

c)

π/6, 5π/6, 7π/6, 11π/6

d)

π/3, 2π/3, 4π/3, 5π/3

23.

Solve on the domain [0, 2π)
2sin x cos x = √2 cos x

a)

π/2, 3π/2

b)

0, π/4, 3π/4

c)

π/2, 3π/2, π/4, 3π/4

d)

No solution

24.

Solve on the domain [0, 2π)
cos² x + sin x + 1 = 0

a)

π

b)

3π/2

c)

π/6, 5π/6, 3π/2

d)

No solution

25.

Solve on the domain [0,2π)
1/4sin x + 1= 0

a)

0

b)

7π/4

c)

5π/4

d)

No solution

26.

Solve equation for $0\le\theta<2\pi$ .
$1+\tan^2\theta=4\tan^2\theta$

a)

$\theta=\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}$

b)

$\theta=\frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3}$

c)

$\theta=\frac{\pi}{6},\frac{7\pi}{6}$

d)

$\theta=\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6}$

27.

Solve equation for $0\le\theta<2\pi$ .
$-3=\cot^2\theta-6$

a)

$\theta=\frac{\pi}{6},\frac{7\pi}{4},\frac{11\pi}{6}$

b)

$\theta=\frac{7\pi}{6},\frac{5\pi}{4},\frac{11\pi}{6}$

c)

$\theta=\frac{\pi}{6},\frac{7\pi}{6}$

d)

$\theta=\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}$

28.

Solve equation for $0\le\theta<2\pi$ .
$-\sin\theta\csc\theta+3\csc\theta=\sqrt{2}\sin\theta+3\csc\theta$

a)

$\theta=\frac{5\pi}{4},\frac{11\pi}{6}$

b)

$\theta=\frac{3\pi}{4},\frac{7\pi}{6},\frac{7\pi}{4}$

c)

$\theta=0,\pi,\frac{5\pi}{4},\frac{7\pi}{4}$

d)

$\theta=\frac{5\pi}{4},\frac{7\pi}{4}$

29.

Solve equation for $0\le\theta<2\pi$ .
$0=-3\cot\theta+2\sqrt{3}\cot\theta\sin\theta$

a)

$\theta=\frac{3\pi}{2}$

b)

$\theta=0,\frac{5\pi}{6},\pi,\frac{11\pi}{6}$

c)

$\theta=0,\frac{\pi}{2},\frac{3\pi}{2}$

d)

$\theta=\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},\frac{3\pi}{2}$

30.

Solve equation for $0\le\theta<2\pi$ .
$\sec^2\theta+3=2\sec\theta+2$

a)

$\theta=0,\frac{\pi}{2},\frac{2\pi}{3},\frac{5\pi}{3}$

b)

$\theta=\frac{\pi}{3},\pi,\frac{5\pi}{3}$

c)

$\theta=\frac{\pi}{4}$

d)

$\theta=0$

31.

Solve equation for $0\le\theta<2\pi$ .
$3\sin^2\theta=7\sin^2\theta+4\sin\theta+1$

a)

$\theta=\frac{7\pi}{6}$

b)

$\theta=0,\frac{\pi}{3},\frac{5\pi}{3}$

c)

$\theta=\frac{3\pi}{4},\frac{7\pi}{4}$

d)

$\theta=\frac{7\pi}{6},\frac{11\pi}{6}$

32.

Solve equation for $0\le\theta<2\pi$ .
$-\sin^2\theta=\sin\theta+1-3\sin^2\theta$

a)

$\theta=\frac{\pi}{2},\frac{11\pi}{6}$

b)

$\theta=\frac{\pi}{2},\frac{7\pi}{6},\frac{11\pi}{6}$

c)

$\theta=\frac{2\pi}{3},\frac{7\pi}{6}$

d)

$\theta=0,\frac{2\pi}{3},\frac{4\pi}{3}$

33.

Solve equation for $0\le\theta<2\pi$ .
$4\cot^2\theta=-1-2\cot\theta+3\cot^2\theta$

a)

$\theta=\frac{3\pi}{4},\frac{7\pi}{4}$

b)

$\theta=\frac{7\pi}{4}$

c)

$\theta=\frac{3\pi}{4}$

d)

$\theta=\frac{\pi}{3},\pi,\frac{5\pi}{3}$

34.

Find ALL solutions.

a)

$\frac{\pi}{3}+2n\pi,\ \pi\ +2n\pi,\ \frac{5\pi}{3}+2n\pi$

b)

$\frac{\pi}{6}+n\pi,\ \pi+n\pi,\ \frac{5\pi}{6}+n\pi$

c)

$\frac{\pi}{6}+2n\pi,\ \pi+2n\pi,\ \frac{5\pi}{6}+2n\pi$

d)

$\frac{2\pi}{3}+2n\pi,\ \pi+2n\pi,\ \frac{7\pi}{3}+2n\pi$

e)

$\frac{7\pi}{6}+n\pi,\ \pi+n\pi,\ \frac{11\pi}{6}+n\pi$

35.

Find all solutions in $\left[0,2\pi\right)$

a)

$\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}$

b)

$0,\ \frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{3}$

c)

$0,\frac{\pi}{4},\frac{3\pi}{4},\frac{7\pi}{4}$

d)

$0,\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6}$

e)

$\frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{6},\frac{11\pi}{6}$

36.

Final ALL solutions.

a)

$\frac{7\pi}{6}+2n\pi,\ \frac{11\pi}{6}+2n\pi,\ \frac{3\pi}{2}+2n\pi$

b)

$\frac{\pi}{6}+2n\pi,\ \frac{5\pi}{6}+2n\pi,\ \frac{3\pi}{2}+2n\pi$

c)

$\frac{7\pi}{6\ }+2n\pi,\ \frac{11\pi}{6}+2n\pi,\ \frac{\pi}{2}+2n\pi$

d)

$\frac{\pi}{6}+2n\pi,\ \frac{5\pi}{6}+2n\pi,\ \frac{\pi}{2}+2n\pi$

37.

Find solutions in $\left[0,2\pi\right)$ .

a)

0

b)

$0,\ \frac{3\pi}{2}$

c)

$0,\ \frac{\pi}{2}$

d)

$0,\ \frac{\pi}{6}$

e)

$0,\frac{11\pi}{6}$

38.

Find ALL solutions.

a)

$\frac{\pi}{2}+4n\pi,\ \frac{7\pi}{2}+4n\pi$

b)

$\frac{3\pi}{2}+4n\pi,\ \frac{5\pi}{2}+4n\pi$

c)

$\frac{\pi}{2}+2n\pi,\ \frac{7\pi}{2}+2n\pi$

d)

$\frac{3\pi}{2}+2n\pi,\ \frac{5\pi}{2}+2n\pi$

39.

Find ALL solutions.

a)

$\frac{\pi}{12}+\frac{n\pi}{3}$

b)

$\frac{\pi}{4}+\frac{n\pi}{2}$

c)

$\frac{5\pi}{12}+\frac{n\pi}{3}$

d)

$\frac{5\pi}{12}+\frac{n\pi}{2}$

40.

Find ALL solutions.

a)

$\frac{\pi}{3}+2n\pi$

b)

$\frac{\pi}{6}+n\pi$

c)

$\frac{\pi}{3}+n\pi$

d)

$\frac{\pi}{6}+2n\pi$

41.

Expand $\sin\ \left(33^{o\ }+42^o\right)$

a)

$\sin\ 75^{o\ }$

b)

$\sin\ 33^o\cos42^o+\cos33^o\sin42^o$

c)

$\sin\ 33^o\cos42^o-\cos33^o\sin42^o$

d)

$\cos33^o\cos42^o+\sin33^o\sin42^o$

42.

Expand $\cos\ \left(\frac{\pi}{5}+\frac{\pi}{6}\right)$

a)

$\cos\frac{\pi}{5}\cos\frac{\pi}{6}-\sin\frac{\pi}{5}\sin\frac{\pi}{6}\$

b)

$\cos\frac{\pi}{5}\cos\frac{\pi}{6}+\sin\frac{\pi}{5}\sin\frac{\pi}{6}\$

c)

$\cos\ \frac{2\pi}{11}$

d)

$\cos\frac{\pi}{5}\sin\frac{\pi}{6}-\cos\frac{\pi}{5}\sin\frac{\pi}{6}\$

43.

$\cos75^o\cos15^o-\sin75^{o\ }\sin15^o$ is equivalent to

a)

$\sin\ 90^{o\ }$

b)

$\sin\ 60^{o\ }$

c)

$\cos\ 90^{o\ }$

d)

$\cos\ 60^{o\ }$

44.

$\frac{\tan45^o+\tan30^o}{1-\tan45^o\tan30^o}$ is equivalent to

a)

$\tan75^o$

b)

$\tan\ 15^o$

c)

$\frac{\sin45^o}{\cos30^o}$

d)

$\tan90^{o\ }$

45.

Use sum or difference angles identity to find the exact value for $\cos105^o$

a)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{-\sqrt{6}-\sqrt{2}}{4}$

d)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

46.

Use sum or difference angles identity to find the exact value for $\sin\ \left(-15^o\right)$

a)

$\frac{\sqrt{6}+\sqrt{2}}{4}$

b)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

c)

$\frac{\sqrt{2}-\sqrt{6}}{4}$

d)

$-\frac{\sqrt{3}}{2}$

47.

Given $\sin x=\frac{3}{5}and\ \sin y=\frac{2}{3},$ where $x\ and\ y$ are both in first quadrant. $Evaluate\ \sin\left(x+y\right)$

a)

$\frac{3\sqrt{5}+8}{15}$

b)

$\frac{4\sqrt{5}+6}{15}$

c)

$\frac{2\sqrt{5}+12}{15}$

d)

$\frac{4\sqrt{5}+2}{5}$

48.

Use a double-angle or half-angle identity to find the exact value of each expression

cos θ = 4/5 and 270° < θ < 360°Find sin 2θ

a)

-1/5

b)

24/25

c)

-24/25

d)

-25/24

49.

Use a double-angle or half-angle identity to find the exact value of each expression
sin θ = −3/5 and 3π/2 < θ < 2π Find tan θ/2

a)

-1/3

b)

-√3

c)

√3/3

d)

-1

50.

Use a double-angle or half-angle identity to find the exact value of each expression
sin θ = −3/5 and 3π/2 < θ < 2π Find tan θ/2

a)

-1/3

b)

-√3

c)

√3/3

d)

-1

51.

Find the exact value of sin 2x if sin x = 12/13 and x is in the first quadrant.

a)

120/169

b)

25/169

c)

60/169

d)

5/13

52.

Use the information about the angle θ, 0≤θ<2π to find the exact value of
sin (2θ)
1. sin θ= 3/5 , 0<θ<π/2

a)

24/7

b)

7/25

c)

24/25

d)

1/2

53.

sin2u =

a)

(sinu)(cosu)

b)

2(sinu)(cosu)

c)

(sinu)^2

d)

2(sinu)^2

54.

Use a half-angle identity to find the exact value of $\sin165\degree$

a)

$\frac{\sqrt{6}-\sqrt{2}}{4}$

b)

$-2-\sqrt{3}$

c)

$\frac{\sqrt{2-\sqrt{2}}}{2}$

d)

$-\frac{\sqrt{3}}{3}$

55.

Use a half-angle identity to find the exact value of $\tan\left(\frac{5\pi}{8}\right)$

a)

$-\sqrt{3+2\sqrt{2}}$

b)

$\frac{\sqrt{2-\sqrt{2}}}{2}$

c)

$-2$

d)

$\frac{\sqrt{3}}{3}$

56.

Use a half-angle identity to find the exact value of $\cot\left(\frac{7\pi}{12}\right)$

a)

$-\frac{\sqrt{3}}{3}$

b)

$-\sqrt{3}$

c)

$\sqrt{4-2\sqrt{2}}$

d)

$-2+\sqrt{3}$

57.

Use a half-angle identity to find the exact value of $\sec67.5\degree$

a)

$0$

b)

$\sqrt{6}+\sqrt{2}$

c)

$\sqrt{4+2\sqrt{2}}$

d)

$1$

58.

$\tan\theta=-\sqrt{14}$ and $\frac{\pi}{2}<\theta<\pi$ find $\cos\left(\frac{\theta}{2}\right)$

a)

$\sqrt{26}$

b)

$\frac{\sqrt{58+10\sqrt{29}}}{2}$

c)

$\frac{\sqrt{450-30\sqrt{15}}}{30}$

d)

$\frac{\sqrt{105-7\sqrt{15}}}{7}$

59.

Find the exact value of the expression: $\sin165\cdot\sin15$

a)

$\frac{\sqrt{3}+2}{2}$

b)

$\frac{-\sqrt{3}+2}{4}$

c)

$\frac{\sqrt{3}-2}{2}$

d)

$\frac{-\sqrt{3}-2}{4}$

60.

Find the exact value of the expression: $\frac{4}{3}\left[\sin255+\sin15\right]$

a)

$\frac{-2\sqrt{3}}{2}$

b)

$\frac{2}{3}$

c)

$\frac{2\sqrt{2}}{3}$

d)

$-\frac{2\sqrt{2}}{3}$

61.

$\cos57\degree\sin55\degree$

a)

$\frac{\cos112\degree-\cos2\degree}{2}$

b)

$\frac{\sin112\degree-\sin2\degree}{2}$

c)

$\frac{\sin2\degree+\sin112\degree}{2}$

d)

$\frac{\sin112\degree-\cos\ 2\degree}{2}$

62.

$\cos129\degree+\cos45\degree$

a)

$-2\cos87\degree\cos42\degree$

b)

$-2\sin87\degree\sin42\degree$

c)

$-2\sin87\degree\cos42\degree$

d)

$2\cos87\degree\cos42\degree$

Trig Identities and Equations

Total questions: 62

Worksheet time: 5hrs 10mins

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