SOLUTION: Solve the equation sin⁡θ = 1−12cos⁡θ for all positive values of θ less than 360∘. Give the answers to three significant digits in the orde

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Question 1068407: Solve the equation sin⁡θ = 1−12cos⁡θ for all positive values of θ less than 360∘. Give the answers to three significant digits in the order of increasing.
Answer by htmentor(1343) About Me  (Show Source):
You can put this solution on YOUR website!
For simplicity in notation, I will subsitute x for theta.
We need to solve sin(x) = 1 - 12*cos(x) for x>=0, x<2*pi
If we use the identity cos^2(x) = 1 - sin^2(x), we can write the expression as:
sin(x) = 1 - 12*sqrt(1-sin^2(x)) -> 12*sqrt(1-sin^2(x)) = 1 - sin(x)
Squaring both sides, we have:
144*(1-sin^2(x)) = 1 - 2*sin(x) + sin^2(x)
Simplify and collect like terms:
145*sin^2(x) - 2*sin(x) - 143 = 0
So we are left with a quadratic in sin(x). Using the quadratic formula gives sin(x)=1, and sin(x)=-0.98621.
sin(x) = 1 -> x = pi/2, or 90 deg.
sin(x) = -0.98621 gives x = -1.4045 rad < 0. The sin function repeats every 2 pi. x+2pi = -1.4045 + 2pi = 4.8787 rad, or 279.5 deg.