Question

If $\sin x = \cos y$ and angle $x$ and angle $y$ are acute. Then what is the relation between $x$ and $y$?
A.$x - y = \dfrac{\pi }{2}$
B.$x + y = \dfrac{{3\pi }}{2}$
C.$x + y = \dfrac{\pi }{2}$
D.$x - y = \dfrac{{3\pi }}{2}$

Answer 1

Here, we will use the trigonometric identities such that we will either convert either side of the given equation into sine function or cosine function. We will solve it further to find the relation between the given angles.

Formula Used:
Trigonometric Identity: $\cos y = \sin \left( {90^\circ - y} \right)$

Complete step-by-step answer:
We are given that $\sin x = \cos y$.
By using the trigonometric identity $\cos y = \sin \left( {90^\circ - y} \right)$, we get
$\Rightarrow \sin x = \sin \left( {90^\circ - y} \right)$
By cancelling the sine function, we get
$\Rightarrow x = 90^\circ - y$
By rewriting the equation, we get
$\Rightarrow x + y = 90^\circ$
$\Rightarrow x + y = \dfrac{\pi }{2}$
Therefore, the relation between $x$ and $y$ is $x + y = \dfrac{\pi }{2}$.
Thus, option (C) is the correct answer.

Additional Information:
An acute angle is an angle made by the two line segments which are less than 90 degrees. An obtuse is an angle made by the two line segments which are greater than 90 degrees and lesser than 180 degrees. A right angle is an angle made by the two line segments which are equal to 90 degrees. A straight angle is an angle made by the two line segments which are equal to 180 degrees. A reflex angle is an angle made by the line segments which are greater than 180 degrees and lesser than 360 degrees. A complete angle is an angle made by the line segments which are equal to 360 degrees.

Note: We know that the sum of two acute angles is always less than or equal to 90 degrees. Sum of two right angles is always a straight angle, sum of two obtuse angles is always a reflex angle. We know that trigonometric identity is an equation that is true. The trigonometric identity used here is the trigonometric relations between the co-ratios of the angles.