Math / Trigonometry

QuestionRewrite $5 \cos 4 x$ in terms of $\cos x$ . $\square$

Studdy Solution

STEP 1

1. We need to express $5 \cos 4x$ using powers of $\cos x$ .
2. We will use multiple angle formulas and identities to achieve this.

STEP 2

1. Use the multiple angle formula for $\cos 4x$ .
2. Express $\cos 4x$ in terms of $\cos 2x$ .
3. Express $\cos 2x$ in terms of $\cos x$ .
4. Simplify the expression.

STEP 3

Use the multiple angle formula for $\cos 4x$ :
$\cos 4x = 2 \cos^2 2x - 1$

STEP 4

Express $\cos 2x$ in terms of $\cos x$ using the double angle formula:
$\cos 2x = 2 \cos^2 x - 1$

STEP 5

Substitute $\cos 2x = 2 \cos^2 x - 1$ into the expression for $\cos 4x$ :
$\cos 4x = 2(2 \cos^2 x - 1)^2 - 1$

STEP 6

Expand the expression $(2 \cos^2 x - 1)^2$ :
$(2 \cos^2 x - 1)^2 = 4 \cos^4 x - 4 \cos^2 x + 1$

STEP 7

Substitute the expanded form back into the expression for $\cos 4x$ :
$\cos 4x = 2(4 \cos^4 x - 4 \cos^2 x + 1) - 1$

STEP 8

Simplify the expression:
$\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 2 - 1$ $\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 1$

STEP 9

Multiply the entire expression by 5 to rewrite $5 \cos 4x$ :
$5 \cos 4x = 5(8 \cos^4 x - 8 \cos^2 x + 1)$ $5 \cos 4x = 40 \cos^4 x - 40 \cos^2 x + 5$
The expression $5 \cos 4x$ in terms of $\cos x$ is:
$\boxed{40 \cos^4 x - 40 \cos^2 x + 5}$

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QuestionRewrite $5 \cos 4 x$ in terms of $\cos x$ . $\square$

Studdy Solution

STEP 1

1. We need to express $5 \cos 4x$ using powers of $\cos x$ .
2. We will use multiple angle formulas and identities to achieve this.

STEP 2

1. Use the multiple angle formula for $\cos 4x$ .
2. Express $\cos 4x$ in terms of $\cos 2x$ .
3. Express $\cos 2x$ in terms of $\cos x$ .
4. Simplify the expression.

STEP 3

Use the multiple angle formula for $\cos 4x$ :
$\cos 4x = 2 \cos^2 2x - 1$

STEP 4

Express $\cos 2x$ in terms of $\cos x$ using the double angle formula:
$\cos 2x = 2 \cos^2 x - 1$

STEP 5

Substitute $\cos 2x = 2 \cos^2 x - 1$ into the expression for $\cos 4x$ :
$\cos 4x = 2(2 \cos^2 x - 1)^2 - 1$

STEP 6

Expand the expression $(2 \cos^2 x - 1)^2$ :
$(2 \cos^2 x - 1)^2 = 4 \cos^4 x - 4 \cos^2 x + 1$

STEP 7

Substitute the expanded form back into the expression for $\cos 4x$ :
$\cos 4x = 2(4 \cos^4 x - 4 \cos^2 x + 1) - 1$

STEP 8

Simplify the expression:
$\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 2 - 1$ $\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 1$

STEP 9

Multiply the entire expression by 5 to rewrite $5 \cos 4x$ :
$5 \cos 4x = 5(8 \cos^4 x - 8 \cos^2 x + 1)$ $5 \cos 4x = 40 \cos^4 x - 40 \cos^2 x + 5$
The expression $5 \cos 4x$ in terms of $\cos x$ is:
$\boxed{40 \cos^4 x - 40 \cos^2 x + 5}$

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QuestionRewrite 5cos⁡4x5 \cos 4 x5cos4x in terms of cos⁡x\cos xcosx. □\square□

Studdy Solution

STEP 1

1. We need to express 5cos⁡4x 5 \cos 4x 5cos4x using powers of cos⁡x \cos x cosx.2. We will use multiple angle formulas and identities to achieve this.

STEP 2

1. Use the multiple angle formula for cos⁡4x \cos 4x cos4x.2. Express cos⁡4x \cos 4x cos4x in terms of cos⁡2x \cos 2x cos2x.3. Express cos⁡2x \cos 2x cos2x in terms of cos⁡x \cos x cosx.4. Simplify the expression.

STEP 3

Use the multiple angle formula for cos⁡4x \cos 4x cos4x:cos⁡4x=2cos⁡22x−1 \cos 4x = 2 \cos^2 2x - 1 cos4x=2cos22x−1

STEP 4

Express cos⁡2x \cos 2x cos2x in terms of cos⁡x \cos x cosx using the double angle formula:cos⁡2x=2cos⁡2x−1 \cos 2x = 2 \cos^2 x - 1 cos2x=2cos2x−1

STEP 5

Substitute cos⁡2x=2cos⁡2x−1 \cos 2x = 2 \cos^2 x - 1 cos2x=2cos2x−1 into the expression for cos⁡4x \cos 4x cos4x:cos⁡4x=2(2cos⁡2x−1)2−1 \cos 4x = 2(2 \cos^2 x - 1)^2 - 1 cos4x=2(2cos2x−1)2−1

STEP 6

Expand the expression (2cos⁡2x−1)2 (2 \cos^2 x - 1)^2 (2cos2x−1)2:(2cos⁡2x−1)2=4cos⁡4x−4cos⁡2x+1 (2 \cos^2 x - 1)^2 = 4 \cos^4 x - 4 \cos^2 x + 1 (2cos2x−1)2=4cos4x−4cos2x+1

STEP 7

Substitute the expanded form back into the expression for cos⁡4x \cos 4x cos4x:cos⁡4x=2(4cos⁡4x−4cos⁡2x+1)−1 \cos 4x = 2(4 \cos^4 x - 4 \cos^2 x + 1) - 1 cos4x=2(4cos4x−4cos2x+1)−1

STEP 8

Simplify the expression:cos⁡4x=8cos⁡4x−8cos⁡2x+2−1 \cos 4x = 8 \cos^4 x - 8 \cos^2 x + 2 - 1 cos4x=8cos4x−8cos2x+2−1 cos⁡4x=8cos⁡4x−8cos⁡2x+1 \cos 4x = 8 \cos^4 x - 8 \cos^2 x + 1 cos4x=8cos4x−8cos2x+1

STEP 9

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QuestionRewrite $5 \cos 4 x$ in terms of $\cos x$ . $\square$

1. We need to express $5 \cos 4x$ using powers of $\cos x$ .
2. We will use multiple angle formulas and identities to achieve this.

1. Use the multiple angle formula for $\cos 4x$ .
2. Express $\cos 4x$ in terms of $\cos 2x$ .
3. Express $\cos 2x$ in terms of $\cos x$ .
4. Simplify the expression.

Use the multiple angle formula for $\cos 4x$ :
$\cos 4x = 2 \cos^2 2x - 1$

Express $\cos 2x$ in terms of $\cos x$ using the double angle formula:
$\cos 2x = 2 \cos^2 x - 1$

Substitute $\cos 2x = 2 \cos^2 x - 1$ into the expression for $\cos 4x$ :
$\cos 4x = 2(2 \cos^2 x - 1)^2 - 1$

Expand the expression $(2 \cos^2 x - 1)^2$ :
$(2 \cos^2 x - 1)^2 = 4 \cos^4 x - 4 \cos^2 x + 1$

Substitute the expanded form back into the expression for $\cos 4x$ :
$\cos 4x = 2(4 \cos^4 x - 4 \cos^2 x + 1) - 1$

Simplify the expression:
$\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 2 - 1$ $\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 1$