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Question: If \( \sin x = \cos x \) and x is acute state the value of x in degrees...

If sin⁡x=cos⁡x\sin x = \cos x and x is acute state the value of x in degrees

Explanation

Solution

Hint : We have been given a trigonometric equation, by making certain changes to it and using various trigonometric identities, we can easily obtain the required value of angle x. This angle x is given to be acute, which means its value is less than 90°.
Trigonometric identities to be used:
sin⁡2x+cos⁡2x=1 sin⁡2x=2sin⁡xcos⁡x   {\sin ^2}x + {\cos ^2}x = 1 \\\ \sin 2x = 2\sin x\cos x \;

Complete step-by-step answer :
We have been given an equation:
sin⁡x=cos⁡x\sin x = \cos x and we need to find the value of angle x in degrees.
This equation can be written as:
sin⁡x−cos⁡x=0\sin x - \cos x = 0
Squaring both the sides, we get:
(sin⁡x−cos⁡x)2=0 sin⁡2x+cos⁡2x−2sin⁡xcos⁡x=0 1−2sin⁡xcos⁡x=0(∵sin⁡2x+cos⁡2x=1)  1=2sin⁡xcos⁡x   {\left( {\sin x - \cos x} \right)^2} = 0 \\\ {\sin ^2}x + {\cos ^2}x - 2\sin x\cos x = 0 \\\ 1 - 2\sin x\cos x = 0\left( {\because {{\sin }^2}x + {{\cos }^2}x = 1} \right) \; 1 = 2\sin x\cos x \;
The value of double sine angle is given as:
sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2\sin x\cos x
Substituting this value, we get:
sin⁡2x=1\sin 2x = 1
The value of sin 90° is 1, so the above equation can be written as:
sin⁡2x=sin⁡90∘ ⇒2x=90∘ ∴x=45∘    \sin 2x = \sin {90^\circ } \\\ \Rightarrow 2x = {90^\circ } \\\ \therefore x = {45^\circ } \;\
It is given that the angle is acute i.e. less than 90° which is also true for the angle obtained.
Therefore, if sin⁡x=cos⁡x\sin x = \cos x and x is acute then the value of x is 45 degrees
So, the correct answer is “ 45° ”.

Note : We can also find the value of angle x by the following method, using the basic formula for tanx i.e. sin⁡xcos⁡x=tan⁡x\dfrac{{\sin x}}{{\cos x}} = \tan x
Given equation: sin⁡x=cos⁡x\sin x = \cos x
Dividing both the sides by cos⁡x\cos x , we get:

sin⁡xcos⁡x=cos⁡xcos⁡x tan⁡x=1 (∵sin⁡xcos⁡x=tan⁡x)  \dfrac{{\sin x}}{{\cos x}} = \dfrac{{\cos x}}{{\cos x}} \\\ \tan x = 1 \\\ \left( {\because \dfrac{{\sin x}}{{\cos x}} = \tan x} \right) \\\

The value of tan is 1 when the angle is equal to 45°, x can be calculated mathematically as:
tan⁡x=1 ⇒x=tan⁡−1(1) ∴x=45∘(∵tan⁡45∘=1)   \tan x = 1 \\\ \Rightarrow x = {\tan ^{ - 1}}(1) \\\ \therefore x = {45^\circ }\left( {\because \tan {{45}^\circ } = 1} \right) \;
Thus, we get the value of x as 45° by following every method.
The angles less than 90° are called acute angles and greater than that are called obtuse.