Trigonometry Examples

$\frac{\sin (3 A)}{\sin (2 A)}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the sine triple-angle identity.

$\frac{- 4 \sin^{3} (A) + 3 \sin (A)}{\sin (2 A)}$

Step 1.2

Factor $\sin (A)$ out of $- 4 \sin^{3} (A) + 3 \sin (A)$ .

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Step 1.2.1

Factor $\sin (A)$ out of $- 4 \sin^{3} (A)$ .

$\frac{\sin (A) (- 4 \sin^{2} (A)) + 3 \sin (A)}{\sin (2 A)}$

Step 1.2.2

Factor $\sin (A)$ out of $3 \sin (A)$ .

$\frac{\sin (A) (- 4 \sin^{2} (A)) + \sin (A) \cdot 3}{\sin (2 A)}$

Step 1.2.3

Factor $\sin (A)$ out of $\sin (A) (- 4 \sin^{2} (A)) + \sin (A) \cdot 3$ .

$\frac{\sin (A) (- 4 \sin^{2} (A) + 3)}{\sin (2 A)}$

Step 2

Apply the sine double-angle identity.

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

Step 3

Cancel the common factor of $\sin (A)$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Factor $- 1$ out of $- 4 \sin^{2} (A)$ .

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

Step 5

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- (4 \sin^{2} (A)) - 1 (- 3)}{2 \cos (A)}$

Step 6

Factor $- 1$ out of $- (4 \sin^{2} (A)) - 1 (- 3)$ .

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

Step 7

Simplify the expression.

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Step 7.1

Rewrite $- (4 \sin^{2} (A) - 3)$ as $- 1 (4 \sin^{2} (A) - 3)$ .

$\frac{- 1 (4 \sin^{2} (A) - 3)}{2 \cos (A)}$

Step 7.2

Move the negative in front of the fraction.

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