Chapter 7: Q. 72 (page 486)

Establish each identity.
$\cos (θ + k π) = {(- 1)}^{k} \cos θ, k$ any integer

Short Answer

Expert verified

The given equation is identity.

Step by step solution

Step 1. Given information

The given equation is $\cos (θ + k π) = {(- 1)}^{k} \cos θ, k$ any integer

Step 2. Use L.H.S. to solve

Use the sum formula for the cosine function,

$\cos (θ + k π) = \cos θ \cdot \cos (k π) - \sin θ \cdot \sin (k π)$

Step 3. Values of coskπ where k can be any integer.

$\cos 0 = \cos 2 π = \cos 4 π = \cos (- 2 π) = \cos (- 4 π) = 1 cosπ = \cos 3 π = \cos 5 π = \cos (- π) = \cos (- 3 π) = \cos (- 5 π) = - 1$

Hence,

$\cos (k π) = {(- 1)}^{k}$

Step 4. Values of sinkπ where k can be any integer.

$\sin 0 = sinπ = \sin 2 π = \sin (- π) = \sin (- 2 π) = 0$

Hence,

$\sin (k π) = 0$

Step 5. Rewrite the expression

$\cos (θ + k π) = \cos θ \cdot \cos (k π) - \sin θ \cdot \sin (k π) = \cos θ \cdot {(- 1)}^{k} - \sin θ \cdot 0 = {(- 1)}^{k} \cdot \cos θ$

Since $L . H . S . = R . H . S .$ , the given equation is identity.

Therefore, the given identity is established.

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Establish each identity.
$\cos (θ + k π) = {(- 1)}^{k} \cos θ, k$ any integer

Short Answer

Step by step solution

Step 1. Given information

Step 2. Use L.H.S. to solve

Step 3. Values of coskπ where k can be any integer.

Step 4. Values of sinkπ where k can be any integer.

Step 5. Rewrite the expression

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Establish each identity.cosθ+kπ=(-1)kcosθ,kany integer

Short Answer

Step by step solution

Step 1. Given information

Step 2. Use L.H.S. to solve

Step 3. Values of coskπ where k can be any integer.

Step 4. Values of sinkπ where k can be any integer.

Step 5. Rewrite the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Discrete Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

Establish each identity.
$\cos (θ + k π) = {(- 1)}^{k} \cos θ, k$ any integer