Question

Use the figure below to find the exact values of the double angles.

Answer 1

Right away, we know that

$\cos\beta=\dfrac{16}{20}=\dfrac45$

which means

$\sin\beta=\sqrt{1-\cos^2\beta}=\dfrac35$

Then

$\cos\beta=\cos2\left(\dfrac\beta2\right)=\cos^2\dfrac\beta2-\sin^2\dfrac\beta2$

$\sin\beta=\sin2\left(\dfrac\beta2\right)=2\sin\dfrac\beta2\cos\dfrac\beta2$

Let $x=\cos\dfrac\beta2$ and $y=\sin\dfrac\beta2$. Then

$\begin{cases}\dfrac45=x^2-y^2\\\\\dfrac35=2xy\end{cases}\implies\begin{cases}\cos\dfrac\beta2=\dfrac3{\sqrt{10}}\\\\\sin\dfrac\beta2=\dfrac1{\sqrt{10}}\end{cases}$

from which we find

$\tan\dfrac\beta2=\dfrac{\sin\frac\beta2}{\cos\frac\beta2}=\dfrac13$

You can use the same ideas above to find the trig ratios for $\dfrac\alpha2$, starting from the given fact that $\sin\alpha=\dfrac{16}{20}=\dfrac45$.