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)