5-3 Solving Trigonometric Equations
Solve each equation for all values of x.
2. 5 = sec 2 x + 3
SOLUTION: The period of secant is 2π, so you only need to find solutions on the interval
are ,
,
, and
. Solutions on the interval (–
the general form of the solutions is
+ 2nπ,
,
+ 2nπ,
. The solutions on this interval
), are found by adding integer multiples of 2π. Therefore,
+ 2nπ,
+ 2nπ,
.
4. 4 tan x – 7 = 3 tan x – 6
SOLUTION: The period of tangent is π, so you only need to find solutions on the interval
is
. Solutions on the interval (–
the solutions is
+ nπ,
,
. The only solution on this interval
), are found by adding integer multiples of π. Therefore, the general form of
.
6. 2 – 10 sec x = 4 – 9 sec x
SOLUTION: The period of secant is 2π, so you only need to find solutions on the interval
are
and
. Solutions on the interval (–
general form of the solutions is
+ 2nπ,
,
. The solutions on this interval
), are found by adding integer multiples of 2π. Therefore, the
+ 2nπ,
.
8. 11 = 3 csc2 x + 7
SOLUTION: eSolutions Manual - Powered by Cognero
Page 1
are
and
. Solutions on the interval (–
,
), are found by adding integer multiples of 2π. Therefore, the
5-3 general
Solving
Trigonometric
form
of the solutions is Equations
+ 2nπ,
+ 2nπ,
.
8. 11 = 3 csc2 x + 7
SOLUTION: The period of cosecant is 2π, so you only need to find solutions on the interval
are
,
,
, and
. Solutions on the interval (–
Therefore, the general form of the solutions is
+ 2nπ,
,
. The solutions on this interval
), are found by adding integer multiples of 2π. + 2nπ,
+ 2nπ,
+ 2nπ,
.
10. 9 + sin2 x = 10
SOLUTION: The period of sine is 2π, so you only need to find solutions on the interval
and . Solutions on the interval (–
form of the solutions is
+ 2nπ,
,
. The solutions on this interval are
), are found by adding integer multiples of 2π. Therefore, the general
+ 2nπ,
.
12. 7 cos x = 5 cos x +
SOLUTION: The period of cosine is 2π, so you only need to find solutions on the interval
are
and
. Solutions on the interval (–
general form of the solutions is
+ 2nπ, ,
. The solutions on this interval
), are found by adding integer multiples of 2π. Therefore, the
+ 2nπ,
.
Find all solutions of each equation on [0, 2 ).
14. –2 sin x = –sin x cos x
eSolutions Manual - Powered by Cognero
SOLUTION: Page 2
are
and
. Solutions on the interval (–
,
), are found by adding integer multiples of 2π. Therefore, the
5-3 general
Solving
Trigonometric
form
of the solutions is Equations
+ 2nπ, + 2nπ,
.
Find all solutions of each equation on [0, 2 ).
14. –2 sin x = –sin x cos x
SOLUTION: The equation cos x = 2 has no real solutions since the maximum value the cosine function can obtain is 1. On the
interval [0, 2π), the equation
has solutions 0 and π.
16. csc2 x – csc x + 9 = 11
SOLUTION: On the interval [0, 2π), the equation csc x = –1 has a solution of
and and the equation csc x = 2 has solutions of .
18. 2 sin2 x = sin x + 1
SOLUTION: On the interval [0, 2π), the equation sin x = 1 has a solution of
and and the equation sin x =
has solutions of .
Find all solutions of each equation on the interval [0, 2 ).
26. cos x – 4 = sin x – 4
SOLUTION: eSolutions Manual - Powered by Cognero
Page 3
and the equation sin x =
On the interval [0, 2π), the equation sin x = 1 has a solution of
has solutions of 5-3 Solving
Equations
.
and Trigonometric
Find all solutions of each equation on the interval [0, 2 ).
26. cos x – 4 = sin x – 4
SOLUTION: Therefore, on the interval [0, 2π) the only valid solutions are
and
.
28. cot2 x csc2 x – cot2 x = 9
SOLUTION: eSolutions Manual - Powered by Cognero
Page 4
Therefore, on the interval [0, 2π) the only valid solutions are
2
2
and
.
2
28. cot
x csc Trigonometric
x – cot x = 9
5-3
Solving
Equations
SOLUTION: is undefined. The solutions of
are Check
eSolutions Manual - Powered by Cognero
Page 5
5-3 Solving Trigonometric Equations
On the interval
, there are no real values of x for which
, but
for 30. sec2 x tan2 x + 3 sec2 x – 2 tan2 x = 3
SOLUTION: 2
The square of any real number must be greater than or equal to zero so sec x = –1 has no solutions. Therefore, on
the interval [0, 2π) the only solutions are 0 and π.
eSolutions Manual - Powered by Cognero
Page 6