Chapter 7: Q. 15 (page 505)

Establish the identity
$\sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$

Short Answer

Expert verified

By applying formula of sin(A+B) we get the identity $\sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$

Step by step solution

Step 1. Given information

$\sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$

Step 2. Rewriting the given expression

We know that $3 θ = θ + 2 θ$

and sin(A+B)= sin(A)cos(B) +cos(A)sin(B)

Therefore ,

$\sin 3 θ = \sin (θ + 2 θ) = \sin θ \cos 2 θ + \cos θ \sin 2 θ$

Step 3. Calculations

We know that
$\cos 2 θ = \cos^{2} θ - \sin^{2} θ \sin 2 θ = 2 \sin θ \cos θ S o, \sin 3 θ = \sin θ \cos 2 θ + \cos θ \sin 2 θ \sin 3 θ = \sin θ (\cos^{2} θ - \sin^{2} θ) + \cos θ (2 \sin θ \cos θ) \sin 3 θ = \sin θ \cos^{2} θ - \sin^{3} θ + 2 \sin θ \cos^{2} θ \sin 3 θ = 3 \sin θ \cos^{2} θ - \sin^{3} θ$

Now,
$\cos^{2} θ = 1 - \sin^{2} θ \sin 3 θ = 3 \sin θ (1 - \sin^{2} θ) - \sin^{3} θ \sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$

LHS=RHS

Hence identity is established.

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Establish the identity
$\sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Rewriting the given expression

Step 3. Calculations

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Establish the identity sin3θ=3sinθ-4sin3θ

Short Answer

Step by step solution

Step 1. Given information

Step 2. Rewriting the given expression

Step 3. Calculations

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Discrete Mathematics

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Establish the identity
$\sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$