Answer

Verified

446.4k+ views

**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.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

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

Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

The polyarch xylem is found in case of a Monocot leaf class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Change the following sentences into negative and interrogative class 10 english CBSE

Casparian strips are present in of the root A Epiblema class 12 biology CBSE