Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 7: Q. 71 (page 486)

Establish each identity.
$\sin (胃 + k 蟺) = {(- 1)}^{k} \sin 胃, k$ any integer

Short Answer

Expert verified

The given equation is identity.

Step by step solution

Step 1. Given information

The given equation is $\sin (胃 + k 蟺) = {(- 1)}^{k} \sin 胃, k$ any integer.

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

Use the sum formula for the sine function,

\sin (胃 + k 蟺) = \sin 胃 路 \cos (k 蟺) + \cos 胃 路 \sin (k 蟺)

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

When $k = 1$ ,

role="math" localid="1646548827797" $\cos (k 蟺) = 肠辞蝉蟺 = - 1$

When $k = 2$ ,

role="math" localid="1646549059159" $\cos (k 蟺) = \cos 2 蟺 = {(- 1)}^{2} = 1$

When $k = 3$ ,

role="math" localid="1646549217406" $\cos (k 蟺) = \cos 3 蟺 = {(- 1)}^{3} = - 1$

Hence, $\cos (k 蟺) = {(- 1)}^{k}$

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

When $k = 1$ ,

$\sin (k 蟺) = 蝉颈苍蟺 = 0$

When $k = 2$ ,

$\sin (k 蟺) = \sin 2 蟺 = 0$

Hence, $\sin (k 蟺) = 0$ for any value of $k$ .

Step 5. Rewrite the expression

$\sin (胃 + k 蟺) = \sin 胃路 \cos (k 蟺) + \cos 胃路 \sin (k 蟺) = \sin 胃路 {(- 1)}^{k} + \cos 胃路 0 = {(- 1)}^{k} 路 \sin 胃$

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

Therefore, the given identity is established.

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91影视

Establish each identity.
$\sin (胃 + k 蟺) = {(- 1)}^{k} \sin 胃, k$ 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

One App. One Place for Learning.

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91影视

Establish each identity.sin胃+k蟺=(-1)ksin胃,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

Theoretical and Mathematical Physics

Statistics

Geometry

Probability and Statistics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Establish each identity.
$\sin (胃 + k 蟺) = {(- 1)}^{k} \sin 胃, k$ any integer