interval (–
, ), are found by adding integer
multiples of 2π. Therefore, the general form of the
solutions is
+ 2nπ,
+ 2nπ,
+ 2nπ,
+
5-3 Solving Trigonometric Equations
2nπ,
.
2
Solve each equation for all values of x.
3.
2 = 4 cos
x + 1
1.
5 sin x + 2 = sin x
SOLUTION:
SOLUTION:
The period of sine is 2π, so you only need to find
solutions on the interval
. The solutions on
The period of cosine is 2π, so you only need to find
solutions on the interval
. The solutions on
and
this interval are
. Solutions on the
interval (–
, ), are found by adding integer
this interval are
,
,
, and
. Solutions on
multiples of 2π. Therefore, the general form of the
the interval (–
, ), are found by adding integer
multiples of 2π. Therefore, the general form of the
+ 2nπ,
+ 2nπ,
solutions is
.
solutions is
+ 2nπ,
+ 2nπ,
+ 2nπ,
+
2
2.
5 = sec
x + 3
2nπ,
.
SOLUTION:
4.
4 tan x – 7 = 3 tan x – 6
SOLUTION:
The period of secant is 2π, so you only need to find
The period of tangent is π, so you only need to find
solutions on the interval
. The solutions on
solutions on the interval
. The only solution on
this interval are ,
,
, and
. Solutions on the
this interval is
. Solutions on the interval (–
, ),
interval (–
, ), are found by adding integer
are found by adding integer multiples of π.
multiples of 2π. Therefore, the general form of the
Therefore, the general form of the solutions is
+
solutions is
+ 2nπ,
+ 2nπ,
+ 2nπ,
+
nπ,
.
2nπ,
.
2
2
5.
9 + cot
x = 12
3.
2 = 4 cos
x + 1
SOLUTION:
SOLUTION:
The period of cotangent is π, so you only need to find
solutions on the interval
. The solutions on this
interval are
and
. Solutions on the interval (–
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The period of cosine is 2π, so you only need to find
, ), are found by adding integer multiples of π.
solutions on the interval
. The solutions on