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Math 1330, Precalculus
Does the given x- value represent a solution to the
given trigonometric equation? Answer yes or no.
x
= ; (^) ( ) ( )
2 2 cos x − 7 cos x = 4
x
= ; (^) ( ) ( )
2 3 tan x = 8 tan x − 5
x
= ; (^) ( ) ( ) ( )
3 2 sin x − 4sin x − 2sin x = − 3
x
= − ; (^) ( ) ( )
2 6 cos x − 7 cos x − 2 = 0
For each of the following equations,
(a) Solve the equation for 0 ≤ x < 360
D D
(b) Solve the equation for 0 ≤ x < 2 π.
(c) Find all solutions to the equation, in radians.
5. (^) ( )
cos 2
x = −
6. (^) ( )
sin 2
x =
7. tan (^) ( x ) = − 1 8. cos (^) ( x (^) )= 0 9. sin (^) ( x ) = 1 10. 3 cot (^) ( x (^) )= − 1 11. (^) ( )
sec 3
x = −
12. csc (^) ( x (^) )= − 2 13. 2 cos (^) ( x (^) )= 2 14. sin (^) ( x ) = 2 15. ( ) ( )
2 2sin x − 5sin x − 3 = 0
16. (^) ( ) ( )
2 2 cos x + cos x = 1
17. ( ) ( )
2 cos x = 2 cos x − 1
18. (^) ( ) ( )
2 2sin x = −7 sin x + 4
When solving an equation such as (^) ( )
1 2
sin x = −
for 0 ≤ x < 360
D D , a typical thought process is to first
compute (^) ( )
(^1 ) 2
sin
− to find the reference angle (in this
case, 30
D ) and then use that reference angle to find
solutions in quadrants where the sine value is negative
(in this case, 210
D and 330
D
. In each of the the
following examples of the form sin ( x )= C ,
(a) Use a calculator to find
1 sin C
− to the nearest
tenth of a degree. This represents the
reference angle.
(b) Use the reference angle from part (a) to find
the solutions to the equation for
0 ≤ x < 360
D D .
19. (^) ( )
sin 5
x = −
20. sin ( x ) = −0. 21. sin (^) ( x ) = −0. 22. (^) ( )
sin 5
x = −
The method in Exercises 19-22 can be used for other
trigonometric functions as well. Use a calculator to
solve each of the following equations for 0 ≤ x < 360
D D .
Round answers to the nearest tenth of a degree. If no
solution exists, state “No solution.”
Note: Since calculators do not contain inverse keys for
cosecant, secant, and cotangent, use reciprocal
relationships to rewrite equations in terms of sine, cosine
and tangent. For example, to solve the equation
sec ( x )= − 4 , first rewrite the equation as ( )
1 4
cos x = − and
then proceed to solve the equation.
23. (a) (^) ( )
cos 7
x = − (b) (^) ( )
csc 5
x =
24. (a) (^) tan ( x ) = − 6 (b) (^) sec ( x )= − 7 25. (a) cot (^) ( x (^) )= −2.9 (b) sin (^) ( x ) =5. 26. (a) (^) ( )
csc 4
x = (b) tan (^) ( x ) =2.
Math 1330, Precalculus
Answer the following.
27. The following example is designed to
demonstrate a common error in solving
trigonometric equations. Consider the equation
( ) ( )
2
2 cos x = 3 cos x for 0 ≤ x < 2 π.
(a) Divide both sides of the equation by cos (^) ( x )
and then solve for x.
(b) Move all terms to the left side of the
equation, and then solve for x.
(c) Are the answers in parts (a) and (b) the
same?
(d) Which method is correct, part (a) or (b)?
Why is the other method incorrect?
28. The following example is designed to
demonstrate a common error in solving
trigonometric equations. Consider the equation
( ) ( )
2 tan x = tan x for 0 ≤ x < 360
D D .
(a) Divide both sides of the equation by tan ( x )
and then solve for x.
(b) Move all terms to the left side of the
equation, and then solve for x.
(c) Are the answers in parts (a) and (b) the
same?
(d) Which method is correct, part (a) or (b)?
Why is the other method incorrect?
Solve the following equations for 0 ≤ x < 360
D D
. If no
solution exists, state “No solution.”
29. (^) ( ) ( )
2 2 sin x =sin x
30. (^) ( ) ( )
2 cos x = −cos x
31. ( ) ( )
2 3 tan x = −tan x
32. (^) ( ) ( )
2 2sin x =sin x
33. (^) ( ) ( ) ( )
2 4sin x cos x − cos x = 0
34. (^) ( ) ( ) ( )
2 sin x tan x =sin x
35. ( ) ( ) ( )
3 2 2 cos x = −3cos x −cos x
36. (^) ( ) ( ) ( )
3 2 2sin x + 9sin x =5sin x
The following exercises show a method of solving an
equation of the form:
sin (^) ( Ax + B (^) )= C , for 0 ≤ x < 2 π.
(The same method can be used for the other five
trigonometric functions as well, and can similarly be
applied to intervals other than 0 ≤ x < 2 π .) Answer
the following, using the method described below.
(a) Write the new interval obtained by
multiplying each term in the solution interval
(in this case, 0 ≤ x < 2 π ) by A and then
adding B.
(b) Let u = Ax + B****. Find all solutions to
sin (^) ( u (^) )= C within the interval obtained in
part (a).
(c) For each solution u from part (b), set up and
solve u = Ax + B for x****. These x- values
represent all solutions to the initial equation.
37. ( )
sin 2 2
x =
38. (^) ( )
sin 3 2
x = −
sin 2 2
x
⎜ −^ ⎟= −
sin 4 2
x
41. sin 3( x + π)= 0
sin 2 3 2
⎛ x π⎞
Solve the following, using either the method above or
the method described in the text. If no solution exists,
state “No solution.”
43. 2 cos 2( x (^) )= 2 , for 0 ≤ x < 2 π 44. 3 tan 2( x )+ 1 = 0 , for 0 ≤ x < 360
D D
45. csc 2( x ) = − 2 , for 0 ≤ x < 360
D D
46. sec 2( x )= 2 , for 0 ≤ x < 2 π
