Question

Give all exact solutions over the interval [0°, 360°].
4cos2θ=8sinθcosθ
Select one:
a.
30° + 360°n, 90° + 360°n, 150° + 360°n, 210° + 360°n, 270° + 360°n, 330° + 360°n, where n is any integer.
b.
11.8° + 360°n, 78.2° + 360°n, 191.8° + 360°n, 258.2° + 360°n, where n is any integer.
c.
22.5° + 360°n, 112.5° + 360°n, 202.5° + 360°n, 292.5° + 360°n, where n is any integer
d.
0° + 360°n, 60° + 360°n, 180° + 360°, 300° + 360°n, where n is any integer.

Answer 1

θ = 22.5° , 112.5° , 202.5° , 292.5°  in the interval [0°, 360°] for 4cos2θ=8sinθcosθ  and 22.5° + 360°n, 112.5° + 360°n, 202.5° + 360°n, 292.5° + 360°n, where n is any integer in the interval  (-∞ , ∞)

Given:

• 4cos2θ=8sinθcosθ

To Find:

• All exact solutions over the interval [0°, 360°]

Solution:

4cos2θ = 8sinθcosθ

Step 1 :

Divide both side by 4:

cos2θ = 2sinθcosθ

Step 2 :

Use identity sin2θ = 2sinθcosθ

cos2θ = sin2θ

Step 3 :

Divide both sides by cos2θ  and use identity sinx/cosx = tanx

1 = tan2θ

Step 4 :

Use tan 45° = 1  and generalized solution for tanx = tanα as x = 180°n + α

tan 45°  = tan2θ

=> 2θ = 180°n +  45°

=> θ = 90°n +  22.5°

All the solution over the interval [0°, 360°].

n = 0 , 1 , 2 , 3 will give

22.5° , 112.5° , 202.5° , 292.5°

Correct answer is  22.5° , 112.5° , 202.5° , 292.5°  in the interval [0°, 360°]

Option c) 22.5° + 360°n, 112.5° + 360°n, 202.5° + 360°n, 292.5° + 360°n, where n is any integer  given a generalized solution over interval (-∞ , ∞)

and only for n = 0 gives  in the interval [0°, 360°]