Algebra and Trigonometry

Review Exercises

Algebra and TrigonometryReview Exercises

Solving Trigonometric Equations with Identities

For the following exercises, find all solutions exactly that exist on the interval$[ 0,2π ). [ 0,2π ).$

1.

$csc 2 t=3 csc 2 t=3$

2.

$cos 2 x= 1 4 cos 2 x= 1 4$

3.

$2 sin θ=−1 2 sin θ=−1$

4.

$tan x sin x+sin( −x )=0 tan x sin x+sin( −x )=0$

5.

$9 sin ω−2=4 sin 2 ω 9 sin ω−2=4 sin 2 ω$

6.

$1−2 tan(ω)= tan 2 (ω) 1−2 tan(ω)= tan 2 (ω)$

For the following exercises, use basic identities to simplify the expression.

7.

$sec x cos x+cos x− 1 sec x sec x cos x+cos x− 1 sec x$

8.

$sin 3 x+ cos 2 x sin x sin 3 x+ cos 2 x sin x$

For the following exercises, determine if the given identities are equivalent.

9.

$sin 2 x+ sec 2 x−1= ( 1− cos 2 x )( 1+ cos 2 x ) cos 2 x sin 2 x+ sec 2 x−1= ( 1− cos 2 x )( 1+ cos 2 x ) cos 2 x$

10.

$tan 3 x csc 2 x cot 2 x cos x sin x=1 tan 3 x csc 2 x cot 2 x cos x sin x=1$

Sum and Difference Identities

For the following exercises, find the exact value.

11.

$tan( 7π 12 ) tan( 7π 12 )$

12.

$cos( 25π 12 ) cos( 25π 12 )$

13.

$sin(70°)cos(25°)−cos(70°)sin(25°) sin(70°)cos(25°)−cos(70°)sin(25°)$

14.

$cos(83°)cos(23°)+sin(83°)sin(23°) cos(83°)cos(23°)+sin(83°)sin(23°)$

For the following exercises, prove the identity.

15.

$cos( 4x )−cos( 3x )cosx= sin 2 x−4 cos 2 x sin 2 x cos( 4x )−cos( 3x )cosx= sin 2 x−4 cos 2 x sin 2 x$

16.

$cos(3x)− cos 3 x=−cos x sin 2 x−sin x sin(2x) cos(3x)− cos 3 x=−cos x sin 2 x−sin x sin(2x)$

For the following exercise, simplify the expression.

17.

$tan( 1 2 x )+tan( 1 8 x ) 1−tan( 1 8 x )tan( 1 2 x ) tan( 1 2 x )+tan( 1 8 x ) 1−tan( 1 8 x )tan( 1 2 x )$

For the following exercises, find the exact value.

18.

$cos( sin −1 ( 0 )− cos −1 ( 1 2 ) ) cos( sin −1 ( 0 )− cos −1 ( 1 2 ) )$

19.

$tan( sin −1 ( 0 )+ sin −1 ( 1 2 ) ) tan( sin −1 ( 0 )+ sin −1 ( 1 2 ) )$

Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

20.

Find$sin( 2θ ),cos( 2θ ), sin( 2θ ),cos( 2θ ),$ and$tan( 2θ ) tan( 2θ )$given$cos θ=− 1 3 cos θ=− 1 3$and$θ θ$is in the interval$[ π 2 ,π ]. [ π 2 ,π ].$

21.

Find$sin( 2θ ),cos( 2θ ), sin( 2θ ),cos( 2θ ),$and$tan( 2θ ) tan( 2θ )$given$sec θ=− 5 3 sec θ=− 5 3$and$θ θ$is in the interval$[ π 2 ,π ]. [ π 2 ,π ].$

22.

$sin( 7π 8 ) sin( 7π 8 )$

23.

$sec( 3π 8 ) sec( 3π 8 )$

For the following exercises, use Figure 1 to find the desired quantities.

Figure 1
24.

25.

For the following exercises, prove the identity.

26.

$2cos( 2x ) sin( 2x ) =cot x−tan x 2cos( 2x ) sin( 2x ) =cot x−tan x$

27.

$cot x cos(2x)=−sin(2x)+cot x cot x cos(2x)=−sin(2x)+cot x$

For the following exercises, rewrite the expression with no powers.

28.

$cos 2 x sin 4 (2x) cos 2 x sin 4 (2x)$

29.

$tan 2 x sin 3 x tan 2 x sin 3 x$

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.

$cos( π 3 ) sin( π 4 ) cos( π 3 ) sin( π 4 )$

31.

$2 sin( 2π 3 ) sin( 5π 6 ) 2 sin( 2π 3 ) sin( 5π 6 )$

32.

$2 cos( π 5 ) cos( π 3 ) 2 cos( π 5 ) cos( π 3 )$

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

33.

$sin( π 12 )−sin( 7π 12 ) sin( π 12 )−sin( 7π 12 )$

34.

$cos( 5π 12 )+cos( 7π 12 ) cos( 5π 12 )+cos( 7π 12 )$

For the following exercises, change the functions from a product to a sum or a sum to a product.

35.

$sin(9x)cos(3x) sin(9x)cos(3x)$

36.

$cos(7x)cos(12x) cos(7x)cos(12x)$

37.

$sin(11x)+sin(2x) sin(11x)+sin(2x)$

38.

$cos(6x)+cos(5x) cos(6x)+cos(5x)$

Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval$[ 0,2π ). [ 0,2π ).$

39.

$tan x+1=0 tan x+1=0$

40.

$2 sin(2x)+ 2 =0 2 sin(2x)+ 2 =0$

For the following exercises, find all exact solutions on the interval$[ 0,2π ). [ 0,2π ).$

41.

$2 sin 2 x−sin x=0 2 sin 2 x−sin x=0$

42.

$cos 2 x−cos x−1=0 cos 2 x−cos x−1=0$

43.

$2 sin 2 x+5 sin x+3=0 2 sin 2 x+5 sin x+3=0$

44.

$cos x−5 sin( 2x )=0 cos x−5 sin( 2x )=0$

45.

$1 sec 2 x +2+ sin 2 x+4 cos 2 x=0 1 sec 2 x +2+ sin 2 x+4 cos 2 x=0$

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval$[0,2π). [0,2π).$Round to four decimal places.

46.

$3 cot 2 x+cot x=1 3 cot 2 x+cot x=1$

47.

$csc 2 x−3 csc x−4=0 csc 2 x−3 csc x−4=0$

For the following exercises, graph each side of the equation to find the approximate solutions on the interval$[0,2π). [0,2π).$

48.

$20 cos 2 x+21 cos x+1=0 20 cos 2 x+21 cos x+1=0$

49.

$sec 2 x−2 sec x=15 sec 2 x−2 sec x=15$