Find all the angles between 0° and 360° such that sec y + 5tan y = 3cos y.?

Answer 1

sec y + 5tan y = 3cos y
Multiply both sides of the equation by cos y
1 + 5sin y = 3cos^2 y

Let sin y = u

1 + 5u = 3(1-u^2)
1 + 5u = 3 - 3u^2
3u^2 + 5u - 2 = 0
3u^2 - u + 6u - 2 = 0
u(3u-1)+2(3u-1) = 0
(3u-1)(u+2) = 0
Since sin y = u, |u|<=1
The only valid value for u = 1/3
sin y = 1/3
y = asin(1/3)
= 19.47°, 160.53° (approx.)

Answer 2

secy + 5tany = 3cosy

Multply thru by cosy.

1 + 5siny = 3cos²y

1 + 5siny = 3(1 - sin²y) = 3 - 3sin²y

3sin²y + 5siny - 2 = 0

(3siny - 1)(siny + 2) = 0

siny = 1/3, -2
Buy siny cannot equal -2 so that solution is rejected.

siny = 1/3

y = arcsin(1/3) ≈ 19.47°, 160.53°