Trig Question?

Question

tan(theta)-sin(theta)=4/9

solve for theta

I feel like I should know this but am stuck. I even know the answer from Maple, but can't figure out the process.

Answer 1

Sketch a right triangle, label side opposite theta, a. And side adjacent theta, b.
The triangle can be any size, so let the hypotenuse be 1.
Then by the Pythagorean theorem:
b = √(1 - a^2)
And the original equation becomes:
a/b - a/1 = 4/9
a/√(1 - a^2) - a = 4/9
a/√(1 - a^2) = (4 + 9a)/9
9a = (4 + 9a)√(1 - a^2)
81a^2 = (16 + 72a + 81a^2)(1 - a^2)
-81a^2 = (a^2 - 1)(81a^2 + 72a + 16)
0 = 81a^4 + 72a^3 + 16a^2 - 72a - 16
Put that in your equation solver, and bear in mind that extraneous roots may be introduced when you square both sides.
a = 0.772650
theta = arc sin(0.772650) = 50.5925degrees