Trigonometry Examples

$\sin (4 x) + \sin (2 x)$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $4 x$ .

$\sin (2 (2 x)) + \sin (2 x)$

Step 1.2

Apply the sine double-angle identity.

$2 (2 \sin (x) \cos (x)) \cos (2 x) + \sin (2 x)$

Step 1.3

Multiply $2$ by $2$ .

$4 (\sin (x) \cos (x)) \cos (2 x) + \sin (2 x)$

Step 1.4

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

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $4$ by $1$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + \sin (2 x)$

Step 1.7

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.7.1

Move $\sin^{2} (x)$ .

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

Step 1.7.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 1.7.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.7.2.2

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

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

Step 1.7.3

Add $2$ and $1$ .

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

Step 1.8

Multiply $- 2$ by $4$ .

$4 \sin (x) \cos (x) - 8 \sin^{3} (x) \cos (x) + \sin (2 x)$

Step 1.9

Apply the sine double-angle identity.

$4 \sin (x) \cos (x) - 8 \sin^{3} (x) \cos (x) + 2 \sin (x) \cos (x)$

Step 2

Add $4 \sin (x) \cos (x)$ and $2 \sin (x) \cos (x)$ .

$6 \sin (x) \cos (x) - 8 \sin^{3} (x) \cos (x)$