tmua.co.uk TMUA Mock Paper 1 · Question 18

Let $\theta$ be a real number. Define two infinite series:

S_1(\theta)=\sin^2\theta+\sin^4\theta+\sin^6\theta+\cdots

S_2(\theta)=\cos^2\theta+\cos^4\theta+\cos^6\theta+\cdots.

Both series converge whenever $|\sin\theta|<1$ and $|\cos\theta|<1$ . Suppose that

S_1(\theta)=\frac23\qquad\text{and}\qquad S_2(\theta)=\frac32.

How many distinct real values of $\theta$ are there in the range $-2\pi\le\theta\le2\pi$ which satisfy both equations?

2

4

6

8

10

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