Download Trigonometry Solving Trig Equations – Homework 3 Identity

Trigonometry
Solving Trig Equations – Homework 3
Identity Replacement
Solve the following equations:
1) cos2x = cos x
2) tan 2x = cot x
3)
3 cot x sin x + 2 cos2 x = 0
4) sin2x = cos x
5) 3sin x − cos x = 0
2
2
6) 3cos2x − 5cos x = 1
7) 4 tan x + sin2x = 0
8) cos x tan x − sin x = 0
2
9) cos 2x + sin x = 1
10) 3tan 2 x + 4sec x = −4
Advanced Algebra & Trigonometry
Solving Trig Equations – Part 4
In the next example, none of our identities will help us get the equation into a form we can solve,
so we will square both sides of the equation. NOTE: When we square both sides of an
equation, we subject ourselves to the possibility of extraneous solutions! We will need to check
our solutions before moving on!
Example:
cos x + 1 = sin x
cos x + 1 = sin x
(cos x + 1) = sin x
2
2
cos x + 2cos x + 1 = 1− cos x
2
2
2 cos2 x + 2cos x = 0
2 cosx(cos x + 1) = 0
cosx = 0
cosx = -1
x = cos −1 0
x = cos−1 (−1)
x = 90° + 180k
x = 180° + 360k
Test both sets of solutions:
cos90° + 1 = sin 90°
0 + 1 = 1 (yes)
cos270° + 1 = sin 270°
0 + 1 = −1 (no)
Final solution:
x = 90° + 360 k
cos180° + 1 = sin180°
−1+ 1 = 0
x = 180° + 360 k
Assignment 4 - Solve the following equations:
1)
2)
3)
4)
5)
6)
7)
8)
sin x = cos x
sin x + cos x = 0
2 cos2 x = sin x + 1
2
sin x − 3sin x + 2 = 0
2
3tan x + 4sec x = −4
sin2 x sin x + cos2 x cos x = 1
cos2 x + 3cos x − 1 = 0
2sin x cos x + 4sin x = cos x + 2
(yes)