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Question: Find the degree measure of angle \(\theta \), if \(\sin \theta = \cos (\theta - 45),\) where \(\thet...

Find the degree measure of angle θ\theta , if sin⁡θ=cos⁡(θ−45),\sin \theta = \cos (\theta - 45), where θ\theta and θ−45∘\theta - {45^ \circ } are acute angles.

Explanation

Solution

Hint: Use the formula sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos ({90^ \circ } - \theta ) and then equate angles on both sides.

Complete step-by-step answer:
From the question,
⇒sin⁡θ=cos⁡(θ−45) .....(i)\Rightarrow \sin \theta = \cos (\theta - 45){\text{ }}.....(i)
We know that, sin⁡θ=cos⁡(90−θ),\sin \theta = \cos (90 - \theta ),putting this value in equation (i)(i):

⇒cos⁡(90∘−θ)=cos⁡(θ−45∘), ⇒90∘−θ=θ−45∘, ⇒2θ=135∘, ⇒θ=135∘2=6712∘  \Rightarrow \cos ({90^ \circ } - \theta ) = \cos (\theta - {45^ \circ }), \\\ \Rightarrow {90^ \circ } - \theta = \theta - {45^ \circ }, \\\ \Rightarrow 2\theta = {135^ \circ }, \\\ \Rightarrow \theta = \dfrac{{{{135}^ \circ }}}{2} = 67{\dfrac{1}{2}^ \circ } \\\

Therefore, the value of angle θ\theta is 6712∘.67{\dfrac{1}{2}^ \circ }.

Note: Whenever we need to solve a trigonometric equation, we try to convert both sides of the equation in the same trigonometric ratio so that we can easily compare their angles.