interval (– , ), are found by adding integer
multiples of 2π. Therefore, the general form of the
5-3 Solving Trigonometric Equations
solutions is
+ 2nπ,
+ 2nπ,
.
9. 6 tan2 x – 2 = 4
Solve each equation for all values of x.
2
5. 9 + cot x = 12
SOLUTION: SOLUTION: The period of cotangent is π, so you only need to find
solutions on the interval
. The solutions on this
The period of tangent is π, so you only need to find
solutions on the interval
. The solutions on this
interval are
interval are
,
and
. Solutions on the interval (–
), are found by adding integer multiples of π. Therefore, the general form of the solutions is
nπ,
+ nπ,
+
.
,
. Solutions on the interval (–
), are found by adding integer multiples of π. Therefore, the general form of the solutions is
nπ,
+ nπ,
11. 7 cot x – 7. 3 csc x = 2 csc x +
and
+
.
= 4 cot x
SOLUTION: SOLUTION: The period of cosecant is 2π, so you only need to
find solutions on the interval
. The solutions
on this interval are
and
. Solutions on the
interval (– , ), are found by adding integer
multiples of 2π. Therefore, the general form of the
solutions is
+ 2nπ,
+ 2nπ,
The period of cotangent is π, so you only need to find
solutions on the interval
. The only solution on
this interval is
.
. Solutions on the interval (–
,
),
are found by adding integer multiples of π. 9. 6 tan2 x – 2 = 4
Therefore, the general form of the solutions is
SOLUTION: nπ,
+
.
Find all solutions of each equation on [0, 2 ).
13. sin4 x + 2 sin2 x − 3 = 0
SOLUTION: The period of tangent is π, so you only need to find
solutions on the interval
. The solutions on this
interval are
,
and
. Solutions on the interval (–
), are found by adding integer multiples of π. eSolutions Manual - Powered by Cognero
Therefore, the general form of the solutions is
+
Page 1
are found by adding integer multiples of π. Therefore, the general form of the solutions is
The equations sin x = 2 and sin x = –2 have no real
solutions. On the interval [0, 2π), the equation cot x =
+
5-3 Solving Trigonometric Equations
nπ,
Find all solutions of each equation on [0, 2 ).
4
0 has solutions
.
2
and .
17. cos3 x + cos2 x – cos x = 1
13. sin x + 2 sin x − 3 = 0
SOLUTION: SOLUTION: when x =
On the interval [0, 2π),
when x =
. Since
On the interval [0, 2π), the equation cos x = 1 has a
solution of 0 and the equation cos x = –1 has a
solution of π.
and is not a real Find all solutions of each equation on the
interval [0, 2 ).
yields no number, the equation
additional solutions.
32. 2
15. 4 cot x = cot x sin x
+ cos x = 2
SOLUTION: SOLUTION: The equations sin x = 2 and sin x = –2 have no real
solutions. On the interval [0, 2π), the equation cot x =
0 has solutions
and .
17. cos3 x + cos2 x – cos x = 1
SOLUTION: eSolutions Manual - Powered by Cognero
On the interval [0, 2π), the equation cos x = 1 has a
solution of 0 and the equation cos x = –1 has a
On the interval [0, 2π), cos x =
when x =
when x =
and .
Page 2