If $ 2 \theta+45^{\circ} $ and $ 30^{\circ}-\theta $ are acute angles, find the degree measure of $ \theta $ satisfying $ \sin \left(2 \theta+45^{\circ}\right)=\cos \left(30^{\circ}-\theta\right) $

Given:

\( 2 \theta+45^{\circ} \) and \( 30^{\circ}-\theta \) are acute angles.

\( \sin \left(2 \theta+45^{\circ}\right)=\cos \left(30^{\circ}-\theta\right) \).

To do:

We have to find the degree measure of \( \theta \).

Solution:  

We know that,

$sin\ (90^{\circ}- \theta) = cos\ \theta$

Let us consider LHS,

$\sin \left(2 \theta+45^{\circ}\right)=\sin \left(90^{\circ}-(2\theta+45^{\circ})\right)$

$=\cos (90^{\circ}-45^{\circ}-2\theta)$

$=\cos (45^{\circ}-2\theta)$

Therefore,

$\cos (45^{\circ}-2\theta)=\cos \left(30^{\circ}-\theta\right)$

Comparing on both sides, we get,

$45^{\circ}- 2\theta=30^{\circ}-\theta$

$2\theta-\theta=45^{\circ}-30^{\circ}$

$\theta=15^{\circ}$

The degree measure of \( \theta \) is $15^{\circ}$.

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Updated on: 10-Oct-2022

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