Transcript 5.3 – Solving Trigonometric Equations

In this section, you 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
 Two basic techniques for solving
trigonometric equations are factoring and
applying known identities.
a) Rewrite the equation with a single
trigonometric function.
b) Remember only to factor after you have
set the equation equal to zero.
Collect Like Terms/Take the Square Root:
1) Solve 2tan x  1   3 and find all solutions.
2 tan x  1   3
2 tan x   2
tan x  1
3
7
x
and
between 0,2 
4
4
3
x
n
4
Collect Like Terms/Take the Square Root:
2) Solve sec x  2  0 in the interval 0,2 .
2
sec x  2  0
2
sec x  2
2
sec x   2
(Solve for extraneous roots.)
2
 3 5 7
cos x  
 x ,
, ,
2
4 4 4 4
 3 5 7
x ,
, ,
4 4 4 4
Use identities to solve:
3) Solve sin 2 x  cos 2 x for all solutions.
2
2
sin x  cos x Set the equation to zero tofactor.
sin x  cos x  0
2
2
Substitute in known identities.
sin x  1  sin x   0 Simplify and solve.
2
2
2sin x  1  0
2
Find all solutions for x.
1
sin x   sin x   2  x     n
2
4 2
2
2
Use identities to solve:
4) Solve 2sin 2 x  cos x  1  0 for the interval 0,2 .
2sin 2 x  cos x  1  0
Substitute in known identities.
2 1  cos x   cos x  1  0
Simplify and combine like terms.
2  2cos x  cos x  1  0
Simplify and combine like terms.
 2cos2 x  cos x  1  0
Multiply by  1.
2cos2 x  cos x  1  0
Factor and solve.
2
2
 2cos x 1 cos x 1  0
1
cos x 
and cos x  1
2
 5
x ,
3 3
Find all solutions for x.

5
x   x  , ,
3
3
Factor to solve:
5) Solve sin x  3sin x  4  0 for the interval 0,2 .
4
2
Factor to solve:
sin 4 x  3sin 2 x  4  0
2
2
sin
x

4
sin

 x  1  0
sin x  4
Factor
Solve separately
2
sin x  1
sin x  2
sin x   1
2
x  no solution since sine
oscillates between  1 and 1
Solving of Multiple Angles:
6) Solve 2sin 4 x  3  0 for all solutions.
2sin 4 x  3  0
3
sin 4 x 
2
4x 
x
x

3

12

12
2
4x 
3
and
x
and
 2 n,

6

6
 2 n
Homework
 Page 376-379
 35-47 odd, 49-53 odd, 61-71 odd