Precalculus Examples

$\sin (4 θ)$

Step 1

Factor $2$ out of $4 θ$ .

$\sin (2 (2 θ))$

Step 2

Apply the sine double-angle identity.

$2 (2 \sin (θ) \cos (θ)) \cos (2 θ)$

Step 3

Multiply $2$ by $2$ .

$4 (\sin (θ) \cos (θ)) \cos (2 θ)$

Step 4

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

$4 \sin (θ) \cos (θ) (2 \cos^{2} (θ) - 1)$

Step 5

Apply the distributive property.

$4 \sin (θ) \cos (θ) (2 \cos^{2} (θ)) + 4 \sin (θ) \cos (θ) \cdot - 1$

Step 6

Multiply $\cos (θ)$ by $\cos^{2} (θ)$ by adding the exponents.

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Step 6.1

Move $\cos^{2} (θ)$ .

$4 \sin (θ) (\cos^{2} (θ) \cos (θ)) \cdot 2 + 4 \sin (θ) \cos (θ) \cdot - 1$

Step 6.2

Multiply $\cos^{2} (θ)$ by $\cos (θ)$ .

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Step 6.2.1

Raise $\cos (θ)$ to the power of $1$ .

$4 \sin (θ) (\cos^{2} (θ) \cos^{1} (θ)) \cdot 2 + 4 \sin (θ) \cos (θ) \cdot - 1$

Step 6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 \sin (θ) {\cos (θ)}^{2 + 1} \cdot 2 + 4 \sin (θ) \cos (θ) \cdot - 1$

Step 6.3

Add $2$ and $1$ .

$4 \sin (θ) \cos^{3} (θ) \cdot 2 + 4 \sin (θ) \cos (θ) \cdot - 1$

Step 7

Multiply $- 1$ by $4$ .

$4 \sin (θ) \cos^{3} (θ) \cdot 2 - 4 \sin (θ) \cos (θ)$

Step 8

Multiply $2$ by $4$ .

$8 \sin (θ) \cos^{3} (θ) - 4 \sin (θ) \cos (θ)$