Solve for 𝜃 if 0° ≤ 𝜃 < 360°. (Enter your answers as a comma-separated list.) 3 sin 𝜃 + cos 𝜃 = 3

Answer 1

3sinθ + cosθ = 3; 6sinθ/2cosθ/2 + cos²θ/2 - sin²θ/2 = 3sin²θ/2 + 3cos²θ/2;

6sinθ/2cosθ/2 - 2cos²θ/2 - 4sin²θ/2 = 0;

This is homogenius equation, we can divide by 2cos²θ/2: 2tan²θ/2 - 3tanθ/2 + 1 = 0;

(2tanθ/2 - 1)(tanθ/2 - 1) = 0; tanθ/2 = 1; θ/2 = 45° and θ = 90°.

tanθ/2 = 1/2; θ = 2tan^-1(1/2) = 53.13°.

Answer 2

3sinx + cosx = 3

3sinx -3 = -cosx = -sqr(1-sin^2x) square both sides

9sin^2x - 18sinx + 9 = 1 -sin^2x

10sin^2x - 18sinx + 8 = 0

5sin^2x -9sinx + 4 = 0 use the quadratic formula

sinx = 9/10 + or - (1/10)sqr(81-80) = 9/10 + 1/10 or 9/10-1/10 = 1 or 0.8

sinx = 1 or .8

x = 90 degrees or about 53.1 degrees

x = 53.1, 90 (when multiplying by a variable, it causes an extraneous solution = 126.9. Ignore it)