Question

If sin (θ + 36°) = cos θ, where (θ + 36°) is acute, find θ.

Answer 1

Given: sin(θ + 36°) = cosθ

$$\begin{gathered} \sin \left(\theta +36°\right)=\cos \theta \\ \Rightarrow \cos \left(90°-\left(\theta +36°\right)\right)=\cos \theta \left(\because \sin \theta =\cos \left(90°-\theta \right)\right) \\ \Rightarrow \cos \left(90°-\theta -36°\right)=\cos \theta \\ \Rightarrow 54°-\theta =\theta \\ \Rightarrow \theta +\theta =54° \\ \Rightarrow 2\theta =54° \\ \Rightarrow \theta =\frac{54°}{2} \\ \Rightarrow \theta =27° \\ \mathrm{Hence},\theta =27°. \end{gathered}$$