Download Solving Trigonometric Equations

Solving Trigonometric Equations
MATH 160, Precalculus
J. Robert Buchanan
Department of Mathematics
Fall 2011
J. Robert Buchanan
Solving Trigonometric Equations
Objectives
In this lesson we will learn to:
use standard algebraic techniques to solve trigonometric
equations,
solve trigonometric equations of quadratic type,
solve trigonometric equations involving multiple angles,
use inverse trigonometric functions to solve trigonometric
equations.
J. Robert Buchanan
Solving Trigonometric Equations
Examples
Solve the following trigonometric equations:
2 sin x + 1 = 0
3 cot2 x − 1 = 0
tan2 3x = 3
2 sin2 x + 3 sin x + 1 = 0
cos x + sin x tan x = 2
sin x − 2 = cos x − 2
√
3
sin 2x = −
2
tan2 x − tan x − 2 = 0
J. Robert Buchanan
Solving Trigonometric Equations
Equation (1 of 8)
2 sin x + 1 = 0
sin x
x=
= −
1
2
7π
11π
+ 2nπ or x =
+ 2nπ
6
6
where n is any integer.
J. Robert Buchanan
Solving Trigonometric Equations
Equation (2 of 8)
cot
2Π
y
-1
=
3
3
3 cot2 x − 1 = 0
1
cot2 x =
3
cot
1.0
1
Π
=
3
3
0.5
1
-1.0
= ±√
3
π
2π
x = + nπ or x =
+ nπ
3
3
0.5
-0.5
cot x
1.0
-0.5
cot
J. Robert Buchanan
4Π
1
=
3
-1.0
cot
3
Solving Trigonometric Equations
5Π
-1
=
3
3
x
Equation (3 of 8)
tan2 3x
= 3
√
tan 3x = ± 3
π
2π
3x = + nπ or 3x =
+ nπ
3
3
π nπ
2π nπ
x= +
or x =
+
9
3
9
3
J. Robert Buchanan
Solving Trigonometric Equations
Equation (4 of 8)
2 sin2 x + 3 sin x + 1 = 0
(2 sin x + 1)(sin x + 1) = 0
2 sin x + 1 = 0 or sin x + 1 = 0
1
or sin x = −1
sin x = −
2
7π
3π
x=
+ 2nπ or x =
+ 2nπ
6
2
J. Robert Buchanan
Solving Trigonometric Equations
Equation (5 of 8)
cos x + sin x tan x
sin2 x
cos x +
cos x
2
cos x + sin2 x
= 2
= 2
= 2 cos x
1 = 2 cos x
1
cos x =
2
π
5π
x = + 2nπ or x =
+ 2nπ
3
3
J. Robert Buchanan
Solving Trigonometric Equations
Equation (6 of 8)
y
1.0
tan
Π
4
=1
0.5
sin x − 2 = cos x − 2
sin x
= cos x
tan x
= 1
π
=
+ nπ
4
x
-1.0
0.5
-0.5
-0.5
tan
J. Robert Buchanan
5Π
4
=1
-1.0
Solving Trigonometric Equations
1.0
x
Equation (7 of 8)
√
sin 2x
= −
3
2
4π
5π
+ 2nπ or 2x =
+ 2nπ
3
3
2π
5π
x=
+ nπ or x =
+ nπ
3
6
2x =
J. Robert Buchanan
Solving Trigonometric Equations
Equation (8 of 8)
tan2 x − tan x − 2 = 0
(tan x − 2)(tan x + 1) = 0
tan x − 2 = 0 or tan x + 1 = 0
tan x = 2 or tan x = −1
3π
x = tan−1 (2) + nπ or x =
+ nπ
4
where n is an integer.
J. Robert Buchanan
Solving Trigonometric Equations
Homework
Read Section 5.3.
Exercises: 1, 5, 9, 13, . . . , 69, 73
J. Robert Buchanan
Solving Trigonometric Equations