Trigonometry Examples

$\sin (2 x) \cos (3 x)$

Step 1

Apply the sine double-angle identity.

$2 \sin (x) \cos (x) \cos (3 x)$

Step 2

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

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

Step 3

Apply the distributive property.

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

Step 4

Multiply $\cos (x)$ by $\cos^{3} (x)$ by adding the exponents.

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Step 4.1

Move $\cos^{3} (x)$ .

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

Step 4.2

Multiply $\cos^{3} (x)$ by $\cos (x)$ .

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $3$ and $1$ .

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

Step 5

Multiply $2 \sin (x) \cos (x) (- 3 \cos (x))$ .

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Step 5.1

Multiply $- 3$ by $2$ .

$2 \sin (x) \cos^{4} (x) \cdot 4 - 6 \sin (x) \cos (x) \cos (x)$

Step 5.2

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

$2 \sin (x) \cos^{4} (x) \cdot 4 - 6 \sin (x) (\cos^{1} (x) \cos (x))$

Step 5.3

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

$2 \sin (x) \cos^{4} (x) \cdot 4 - 6 \sin (x) (\cos^{1} (x) \cos^{1} (x))$

Step 5.4

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

$2 \sin (x) \cos^{4} (x) \cdot 4 - 6 \sin (x) {\cos (x)}^{1 + 1}$

Step 5.5

Add $1$ and $1$ .

$2 \sin (x) \cos^{4} (x) \cdot 4 - 6 \sin (x) \cos^{2} (x)$

Step 6

Multiply $4$ by $2$ .

$8 \sin (x) \cos^{4} (x) - 6 \sin (x) \cos^{2} (x)$