Answer

Verified

395.7k+ views

**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 ^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 $ and we need to find the value of angle x in degrees.

This equation can be written as:

$ \sin x - \cos x = 0 $

Squaring both the sides, we get:

$

{\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 = 2\sin x\cos x $

Substituting this value, we get:

$ \sin 2x = 1 $

The value of sin 90° is 1, so the above equation can be written as:

$

\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 $ 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. \[\dfrac{{\sin x}}{{\cos x}} = \tan x\]

Given equation: $ \sin x = \cos x $

Dividing both the sides by $ \cos x $ , we get:

\[

\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 \\

\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.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Define limiting molar conductivity Why does the conductivity class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Name 10 Living and Non living things class 9 biology CBSE

The Buddhist universities of Nalanda and Vikramshila class 7 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE