Math CentralQuandaries & Queries


Question from Bobby, a student:

cos 2x = 2 sin x


To solve this equation for x my first thought is to use the multiple angle expression for the cosine function

cos(A + B) = cos(A) cos(B) - sin(A) sin(B)

In your case A = B = x so you get

cos(2x) = cos2(x) - sin2(x).

Thus your equation becomes

cos2(x) - sin2(x) = 2 sin(x).

This still involves sine functions and cosine functions, but I know that sin2(x) + cos2(x) = 1, or cos2(x) = 1 - sin2(x) so the equation can be written

1 - sin2(x) - sin2(x) = 2 sin(x)


2 sin2(x) + 2 sin(x) - 1 = 0.

Write y = sin(x) and this becomes a quadratic in y. Solve the quadratic for y, write the solutions y1 and y2 in the form sin(x) = y1 and sin(x) = y2 and solve for x. Remember that -1 ≤ sin(x) ≤ 1.

You didn't give any restrictions on x so make sure you have all the solutions.


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