Trigonometry Examples

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

Step 1

Given the expression $a \sin (x) + b \cos (x)$ , find the values of $k$ and $θ$ .

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

$θ = \tan^{-1} (\frac{b}{a})$

Step 2

Calculate the value for $k$ by substituting the coefficients from $2 \sin (x) \cos^{3} (x)$ and $2 \sin^{3} (x) \cos (x)$ into $k = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise $2$ to the power of $2$ .

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

Step 2.2

Raise $2$ to the power of $2$ .

$k = \sqrt{4 + 4}$

Step 2.3

Add $4$ and $4$ .

$k = \sqrt{8}$

Step 2.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.4.1

Factor $4$ out of $8$ .

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

Step 2.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.5

Pull terms out from under the radical.

$k = 2 \sqrt{2}$

Step 3

Find the value for $θ$ by substituting the coefficients from $2 \sin (x) \cos^{3} (x)$ and $2 \sin^{3} (x) \cos (x)$ into $θ = \tan^{-1} (\frac{b}{a})$ .

$θ = \tan^{-1} (\frac{2}{2})$

Step 4

Divide $2$ by $2$ .

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

Step 5

Linear combinations of trigonometric functions dictate that $a \sin (x) + b \cos (x) = k \sin (x + θ)$ . Substitute the values of $k$ and $θ$ .

$2 \sqrt{2} \sin (x + \frac{π}{4})$