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The number of solutions of the equation is $\sin x = \cos 3x$ in $\left[ {0,\pi } \right]$ is
A) 1
B) 2
C) 3
D) 4

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Last updated date: 19th Jun 2024
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Answer
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Hint: For solving trigonometric equations, use both the reference angles and trigonometric identities.
As a general description, there are 3 steps. These steps may be very challenging, depending on the equation.
Step 1: Find the trigonometric values needed to solve the equation.
Step 2: Find all 'angles' that give us these values from step 1.
Step 3: Find the values of the unknown that will result in angles that we got in step 2.

Formula used:
\[
  1{\text{ }} + {\text{ ta}}{{\text{n}}^{\;2}}x{\text{ }} = {\text{ se}}{{\text{c}}^{\;2}}x \\
  cos3x = 4co{s^3}3x - 3cosx \\
 \]

Complete step-by-step answer:
Just as with linear equations, we must first isolate the variable-containing term:
The given equation using trigonometry identity can be written as:-
\[
  sinx = 4co{s^3}x - 3cosx \\
   \Rightarrow \sin x = \dfrac{4}{{{{\sec }^3}x}} - \dfrac{3}{{\sec x}} \\
   \Rightarrow \dfrac{{\sin xse{c^2}x}}{{\cos x}} = 4 - 3{\sec ^2}x \\
   \Rightarrow se{c^2}xtanx + 3se{c^2}x - 4 = 0 \\
   \Rightarrow se{c^2}x(tanx + 3) - 4 = 0 \\
 \]
In terms of tan x, this leads to the equation
\[
   \Rightarrow se{c^2}x(tanx + 3) - 4 = 0 \\
   \Rightarrow (1 + {\tan ^2}x)(tanx + 3) - 4 = 0 \\
   \Rightarrow \tan x + 3 + {\tan ^3}x + 3{\tan ^2}x - 4 = 0 \\
 \]
Let us factorise the left hand side of the equation using simple factorisation and then we have to solve for each of the factors.
\[
   \Rightarrow \tan x + {\tan ^3}x + 3{\tan ^2}x + \tan x - 1 = 0 \\
   \Rightarrow (\tan x + 1)(\tan 2x + 2\tan x - 1) = 0 \\
 \]
After the factorisation, we are left with two trigonometric equations. Now let us further simplify those trigonometric equations separately and find the possible values for the equations.
\[
   \Rightarrow \tan x = - 1\;or\;\tan 2x = 1 \\
   \Rightarrow x = \dfrac{{3\pi }}{4},\dfrac{\pi }{8},\dfrac{{5\pi }}{8} \\
 \]
So, option (C) is the correct answer.

Note: 1. If tan θ or sec θ is involved in the equation then θ ≠ odd multiple of π/2.
2. If cot θ or cosec θ is involved in the equation then θ ≠ multiple of π or 0.
Trigonometry is full of formulas and the students are advised to learn all the trigonometric formulas including the trigonometry basics so as to remain prepared for examination. Students must practice various trigonometry problems based on trigonometric ratios and trigonometry basics so as to get acquainted with the topic.