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stealth61 [152]
3 years ago
5

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

Mathematics
1 answer:
scZoUnD [109]3 years ago
3 0

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.

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