Trigonometry Examples

2 sin (x) {cos}^{3} (x) + 2 {sin}^{3} (x) cos (x)

$2 \sin (x) \cos^{3} (x) + 2 \sin^{3} (x) \cos (x)$

Step 1

Given the expression

a sin (x) + b cos (x)

$a \sin (x) + b \cos (x)$ , find the values of

k

$k$ and

θ

$θ$ .

k = \sqrt{a^{2} + b^{2}}

$k = \sqrt{a^{2} + b^{2}}$

θ = {tan}^{-1} (\frac{b}{a})

$θ = \tan^{-1} (\frac{b}{a})$

Step 2

Calculate the value for

k

$k$ by substituting the coefficients from

2 sin (x) {cos}^{3} (x)

$2 \sin (x) \cos^{3} (x)$ and

2 {sin}^{3} (x) cos (x)

$2 \sin^{3} (x) \cos (x)$ into

k = \sqrt{a^{2} + b^{2}}

$k = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise

2

$2$ to the power of

2

$2$ .

k = \sqrt{4 + {(2)}^{2}}

$k = \sqrt{4 + {(2)}^{2}}$

Step 2.2

Raise

2

$2$ to the power of

2

$2$ .

k = \sqrt{4 + 4}

$k = \sqrt{4 + 4}$

Step 2.3

Add

4

$4$ and

4

$4$ .

k = \sqrt{8}

$k = \sqrt{8}$

Step 2.4

Rewrite

8

$8$ as

2^{2} \cdot 2

$2^{2} \cdot 2$ .

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Step 2.4.1

Factor

4

$4$ out of

8

$8$ .

k = \sqrt{4 (2)}

$k = \sqrt{4 (2)}$

Step 2.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

k = \sqrt{2^{2} \cdot 2}

$k = \sqrt{2^{2} \cdot 2}$

k = \sqrt{2^{2} \cdot 2}

$k = \sqrt{2^{2} \cdot 2}$

Step 2.5

Pull terms out from under the radical.

k = 2 \sqrt{2}

$k = 2 \sqrt{2}$

k = 2 \sqrt{2}

$k = 2 \sqrt{2}$

Step 3

Find the value for

θ

$θ$ by substituting the coefficients from

2 sin (x) {cos}^{3} (x)

$2 \sin (x) \cos^{3} (x)$ and

2 {sin}^{3} (x) cos (x)

$2 \sin^{3} (x) \cos (x)$ into

θ = {tan}^{-1} (\frac{b}{a})

$θ = \tan^{-1} (\frac{b}{a})$ .

θ = {tan}^{-1} (\frac{2}{2})

$θ = \tan^{-1} (\frac{2}{2})$

Step 4

Divide

2

$2$ by

2

$2$ .

{tan}^{-1} (1)

$\tan^{-1} (1)$

Step 5

Linear combinations of trigonometric functions dictate that

a sin (x) + b cos (x) = k sin (x + θ)

$a \sin (x) + b \cos (x) = k \sin (x + θ)$ . Substitute the values of

k

$k$ and

θ

$θ$ .

2 \sqrt{2} sin (x + \frac{π}{4})

$2 \sqrt{2} \sin (x + \frac{π}{4})$