If $\sin( 2 \theta) = \cos( 3)$ and $\theta \leq 90°$, find $\theta$

Question

Find $\theta\leq90°$ if $$\sin( 2 \theta) = \cos( 3)$$

I know that $\sin 2\theta = 2\sin\theta\cos \theta$, or alternatively, $\theta = \dfrac{\sin^{-1}(\cos 3)}{2}$.

Can somebody help me?

Answer 1

Another way to solve the problem: note that

$$\sin(2\theta)=\cos(3)=\sin(90-3)$$

But we know $\theta\leq90°$, so $2\theta\leq180°$. Then, two solutions are possible:

$$2\theta=\left\{\begin{array}{ccc}90-3=87°&\Rightarrow&\theta_1=43.5°\\90+3=93°&\Rightarrow&\theta_2=46.5°\end{array}\right.$$

Answer 2

$$\sin{( 2 \theta)} = \cos{(3^{\circ})} \Rightarrow \sin{(2 \theta)}=\sin{ \left (90^{\circ} \pm 3^{\circ} \right )} \Rightarrow 2 \theta=90^{\circ} \pm 3^{\circ}+360^{\circ}k, k \in \mathbb{Z} \\ \Rightarrow \theta=\left ( 45\pm\frac{3}{2}+180k \right )^{\circ}, k \in \mathbb{Z}$$

$$\Rightarrow \theta=\left ( \frac{93}{2}+180k \right )^{\circ} \text{ or } \theta=\left ( \frac{87}{2}+180k \right )^{\circ}, \ \ \ k\in \mathbb{Z}$$

Then you have to find a $\theta$ such that $\theta \leq 90^{\circ}$.

Answer 3

Assuming you mean $3^{\circ}$, here are the steps $$ \sin 2\theta=\cos 3^{\circ} $$ $$ \arcsin(\sin 2\theta)=\arcsin(\cos 3^{\circ}) $$ $$ 2\theta=\arcsin(\cos 3^{\circ}) $$ $$ \theta=\frac{1}{2}\arcsin(\cos 3^{\circ})=\frac{87^{\circ}}{2}=43.5^{\circ} $$

Answer 4

As you've noted, $\theta = \dfrac{\sin^{-1}(\cos 3)}{2}$. Cosine of $3$ is in the second quadrant and is quite close to $-1$, and sine of something close to $-1$ is going to be in the fourth quadrant, hence this sine is going to be negative. Dividing by two doesn't change the sign, so the aswer is:

$$\theta = -\dfrac{180\sin^{-1}(\cos 3)}{2\pi}=-\dfrac{90\sin^{-1}(\cos 3)}{\pi}$$

(The factor of $\frac{180}{\pi}$ comes from the fact that we need to convert radians to degrees.)

Answer 5

$$\sin2\theta=\cos3^\circ=\sin87^\circ$$

$$\implies2\theta=180^\circ m+(-1)^m87^\circ\text{ where } m \text{ is any integer}$$

$$\implies\theta=90^\circ m+(-1)^m43.5^\circ$$

Check for even $m=2r$(say) and for odd $=2r+1$(say)

Find $m$ such that $\displaystyle\theta\le90^\circ$

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Alternatively, $$\cos3^\circ=\sin2\theta=\cos\left(90^\circ-2\theta\right)=\cos\left(2\theta-90^\circ\right)$$

$$\iff2\theta-90^\circ= 360^\circ n\pm3^\circ\text{ where } n \text{ is any integer}$$

$\displaystyle'+'\implies 2\theta= 360^\circ n+93^\circ\iff\theta=180^\circ n+46.5^\circ$

$\displaystyle'-'\implies 2\theta= 360^\circ n+87^\circ\iff\theta=180^\circ n+43.5^\circ$

Find $n$ such that $\displaystyle\theta\le90^\circ$