
If $ \sin x = \cos x $ and x is acute state the value of x in degrees
Answer
466.2k+ 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.
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
Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Statement I Reactivity of aluminium decreases when class 11 chemistry CBSE

Trending doubts
10 examples of friction in our daily life

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Prokaryotic Cells and Eukaryotic Cells

State and prove Bernoullis theorem class 11 physics CBSE

What organs are located on the left side of your body class 11 biology CBSE

How many valence electrons does nitrogen have class 11 chemistry CBSE
