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If $\cos 2\theta = \sin 4\theta $, where $2\theta $ and $4\theta $ are acute angles, find the value of $\theta $.

Last updated date: 16th Jun 2024
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Hint:As we know that the above given question is related to trigonometric expression, sine and cosine are trigonometric ratios. Here we have to find the value using trigonometric identity or formulae. We know that any angle is acute only if its value is less than ${90^ \circ }$. WE will use this basic formula to find the value of required expression.

Complete step by step solution:
As per the given question we have $\cos 2\theta = \sin 4\theta $, and we have to find the value of $\theta $, also that $2\theta ,4\theta $ are acute angles.
We know the formula of any trigonometric acute angle can be written as : $\cos \theta = \sin (90 - \theta )$. We can use this for $\cos 2\theta $ and it can be written as $\cos 2\theta = \sin (90 - 2\theta )$. So by substituting the value we get: $\sin (90 - 2\theta ) = 1 \Rightarrow \sin 4\theta $.
We can write it as $90 - 2\theta = 4\theta $, as the sine on both sides get cancelled. Now we solve for $\theta $, $90 = 2\theta + 4\theta \Rightarrow 6\theta = 90$.
It gives us $\theta = \dfrac{{90}}{6} = {15^ \circ }$.
Hence the required value of $\theta $ is ${15^ \circ }$.

Note: Before solving such a question we should be fully aware of the trigonometric identities, ratios and their formulas. The important step is to determine the value of $\theta $, and it is given that the value is an acute angle, so our answer should always be less than ${90^ \circ }$. We should remember them as we need to use them in solving questions like this. We should be careful while doing the calculation because if there is mistake in calculation, we might get the wrong answer.