Download Simple Trigonometric Equations

Simple
Trigonometric
Equations
Examples for trigonometric
equations
2 cos x  3 cos x  8  0
sin x  cos x  1
2
tan x  cot x  3
Recall
cos x  sin x  1
2
2
1  tan x  sec x
2
2
1  cot x  csc x
sin x
tan x 
cos x
2
2
sin 
cos 
tan 

sin 
cos 
tan 
180
sin 
-cos 
-tan 
180+
-sin 
-cos 
tan 

180
S
A
T
C
180
360
360
-sin 
cos 
-tan 
Example
Solve the equation cos x=1/2.
Solution:
1
cos x 
2
x  60or 360  60
x  60or 300

360
where  = 60
Example
Solve the equation sin x = -0.7 for
0<x<2.
Solution:
sin x  0.7
x    0.775or 2  0.775
x  3.917or 5.508

2
where   0.775
Class Practice
Solve the following equations for 0360.
1
1. sin x =
2
2. tan x = 1
3. cos x = -0.1
Answers
1. X=30 or 150.
2. X=45  or 225  .
3. X=95.74  or 275.74  .
Procedure in solving
trigonometric equations
Example
Solve the equation
Solution:
2 cos 2 x  3 sin x  0 ,
for 0 x  360
2 cos 2 x  3 sin x  0
2(1  sin 2 x)  3 sin x  0
2 sin 2 x  3 sin x  2  0
(2 sin x  1)(sin x  2)  0
2 sin x  1  0or sin x  2  0
1
sin x   , or sin x  2(rejected )
2
x  180  30, or 360  30
x  210, or 330
180
360
Example
Solve the equation
for 0    360  .
sec 2   1  3 tan 
Solution:
sec 2   1  3 tan 
(tan 2   1)  1  3 tan 

tan 2   3 tan   2  0
(tan   2)(tan   1)  0
tan   2, or tan   1
180
  63.43,180  63.43, or  45, or180  45
  63.43,243.43, or 45,225
Example
Solve the equation 5 cos
for 0  x < .
2
x  4 sin x cos x  2 sin 2 x  2
Solution:
5 cos 2 x  4 sin x cos x  2 sin 2 x  2(sin 2 x  cos 2 x)
3 cos 2 x  4 sin x cos x  4 sin 2 x  0
cos 2 x 4 sin x cos x 4 sin 2 x
3(
)

0
cos 2 x
cos 2 x
cos 2 x
4 tan 2 x  4 tan x  3  0


(2 tan x  3)( 2 tan x  1)  0
3
1
, or tan x  
2
2
x  0.98,   0.98, or  0.46,2  0.46
tan x 
x  0.98,4.12.or 2.68,5.82

2
Class Practice
Textbook
Ex.5C : 1 - 5