Given $\cos (\theta)$ and $\sin (\theta)$, find $2\theta$

Question

I am working on my scholarship exam. I worked through almost final step but got my answer wrong. Could you please have a look?

If $\cos (\theta) = \sqrt{\frac{1}{2}+\frac{1}{2\sqrt{2}}}$ and $\sin (\theta) = -\sqrt{\frac{1}{2}-\frac{1}{2\sqrt{2}}}$ with $0\leq\theta<2\pi$, it follows that $2\theta = ..... \pi$

What I have got is below:

$\sin(2\theta)=2\sin\theta\cos\theta$

Then, $\sin(2\theta)=-\frac{1}{\sqrt{2}}$

Hence, $2\theta = \frac{5\pi}{4}$ or $\frac{7\pi}{4}$ (quadrant 3 or 4)

$\theta=\frac{5\pi}{8}$ or $\frac{7\pi}{8}$

Since $\cos\theta$ is positive and $\sin\theta$ is negative, $\theta$ should be in quadrant 4 but my $\theta$'s are not. So I cannot use my $2\theta$ as a final answer.

However, the answer key provided is $\theta=\frac{15\pi}{4}$, Why do you think that is the case? How did they get to this answer? Please help.

Answer 1

If $0\le \theta<2\pi$, then $0\le 2\theta<4\pi$. You should have $4$ possible values of $\theta$.

$\cos (\theta) = \sqrt{\frac{1}{2}+\frac{1}{2\sqrt{2}}}$ and $\sin (\theta) = -\sqrt{\frac{1}{2}-\frac{1}{2\sqrt{2}}}$ imply that $\sin(2\theta)=-\dfrac1{\sqrt2}$, but not the other way round.

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Note that $\displaystyle \tan\theta=\frac{-\sqrt{\frac{1}{2}-\frac{1}{2\sqrt{2}}}}{\sqrt{\frac{1}{2}+\frac{1}{2\sqrt{2}}}}=-\sqrt{\frac{\sqrt2-1}{\sqrt2+1}}=1-\sqrt{2}$.

So $\theta=n\pi-\dfrac{\pi}{8}$, ($n\in\mathbb{Z}$).

[Note: $\frac{\tan(-\frac\pi8)}{1-\tan^2(-\frac\pi8)}=\tan2(-\frac\pi8)=-1$ implies that $\tan(-\frac\pi8)=1-\sqrt2$.]

As $\theta\in[0,2\pi)$, $\cos\theta>0$ and $\sin\theta<0$, we have $\theta=2\pi-\dfrac{\pi}{8}$ and hence $2\theta=\dfrac{15\pi}4$.

Answer 2

Your mistake is at this step: $\sin(2\theta)=-\frac{1}{\sqrt{2}}$, hence $2\theta = \frac{5\pi}{4}$ or $\frac{7\pi}{4}$. The correct step is: $\sin(2\theta)=-\frac{1}{\sqrt{2}}$, hence $2\theta = \frac{5\pi}{4}+2k\pi$ or $\frac{7\pi}{4}+2k\pi$, where $k$ is any integer. So dividing by 2 we get $\theta = \frac{5\pi}{8}+k\pi$ or $\frac{7\pi}{8}+k\pi$, where $k$ is any integer. If you take $k=0$, the answer doesn't work. So you need to use a different $k$.

Answer 3

Hence, $2\theta = \frac{5\pi}{4}$ or $\frac{7\pi}{4}$

Don't forget $+2\pi n$.

Since the signs of $\cos\theta$ and $\sin\theta$ place $\theta$ in the fourth quadrant, only two candidates hold up. You can choose between them by testing whether $\cos\theta > \cos\frac{7\pi}{4}$

Answer 4

Remember that if the range of $\theta$ is $0 \leq \theta \lt 2 \pi$ then the range of $2\theta$ will be $0\leq\theta \lt 4\pi$. So $2\theta = \ldots$

Answer 5

The given information has $\cos$ positive and $\sin$ negative so $\theta$ is in the 4th quadrant. Makes, $2\theta$ in the 8th quadrant that is $7\pi/4\leq 2\theta \leq 4\pi$; which angle in the 8th quadrant gives $\sin 2\theta = \frac{-1}{\sqrt 2}$? if you want to add more that $ \cos 2\theta = \frac{1}{\sqrt 2}$. How to find angle the 8th quadrant, $\theta$ in 4th quadrant $+ 2\pi$.