$ \cos2\alpha$ and $\sin2\beta$

Hello! I have question from my teacher.

If $\dfrac{\sin \alpha}{\sin\beta} = 3$ and $\dfrac{\cos \alpha}{\sin\beta} = \dfrac13$ , find $\dfrac{\cos2\alpha}{\sin2\beta}$ .

#Geometry

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Substitute the value of $\sin(\beta)$ into the second equation. You get $\cot(\alpha)=\dfrac 1 9$ .

Now draw the triangle with the required ratios.

Now use the double angle identity and expand the numerator and denominator of the desired ratio.

Now simplify the expression from the values you get from the triangle you drew.

Finally you get the answer as $\dfrac{40} 9$ .

A Former Brilliant Member - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

\( \cos2\alpha\) and \(\sin2\beta\)

Comments