Question

If \[\sin \theta =\cos \left( \theta -{{6}^{\circ }} \right)\], find the value of \[\theta \].

Answer 1

Hint: Use the trigonometric formula \[\sin \theta =\cos \left( {{90}^{\circ }}-\theta \right)\] to simplify the given trigonometric expression. Rearrange the terms of the equation and then simplify it to find the value of angle which satisfies the given trigonometric equation.

Complete step-by-step answer:
We have the equation \[\sin \theta =\cos \left( \theta -{{6}^{\circ }} \right)\]. We have to find the value of \[\theta \] which satisfies the given equation.
We know that \[\sin \theta =\cos \left( {{90}^{\circ }}-\theta \right)\].
Substituting the above equation in equation \[\sin \theta =\cos \left( \theta -{{6}^{\circ }} \right)\], we have \[\cos \left( {{90}^{\circ }}-\theta \right)=\cos \left( \theta -{{6}^{\circ }} \right)\].
Thus, we have \[{{90}^{\circ }}-\theta =\theta -{{6}^{\circ }}\].
Rearranging the terms, we have \[2\theta ={{90}^{\circ }}+{{6}^{\circ }}={{96}^{\circ }}\].
Thus, we have \[\theta =\dfrac{{{96}^{\circ }}}{2}={{48}^{\circ }}\].
Hence, the value of \[\theta \] which satisfies the given equation is \[\theta ={{48}^{\circ }}\].

Note: We can also solve this question by using the identity \[\cos \left( x-y \right)=\cos x\cos y+\sin x\sin y\]. Substitute \[x=\theta ,y={{6}^{\circ }}\] in the identity and simplify it to get \[\tan \theta =\dfrac{\cos {{6}^{\circ }}}{1-\sin {{6}^{\circ }}}\]. Substitute the values \[\cos {{6}^{\circ }}=\dfrac{1}{4}\sqrt{7+\sqrt{5}+\sqrt{30+6\sqrt{5}}}\] and \[\sin {{6}^{\circ }}=\dfrac{\sqrt{9-\sqrt{5}-\sqrt{30+6\sqrt{5}}}}{4}\] in the previous equation. Also, use the fact that \[{{\tan }^{-1}}\left( 1.11 \right)={{48}^{\circ }}\] to get the value of \[\theta \] which satisfies the given equation.