Trigonometry Examples

$\sin (a) (3 - 4 \sin^{2} (a)) = \sin (3 a)$

Step 1

Simplify $\sin (a) (3 - 4 \sin^{2} (a))$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$\sin (a) \cdot 3 + \sin (a) (- 4 \sin^{2} (a)) = \sin (3 a)$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $3$ to the left of $\sin (a)$ .

$3 \cdot \sin (a) + \sin (a) (- 4 \sin^{2} (a)) = \sin (3 a)$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$3 \cdot \sin (a) - 4 \sin (a) \sin^{2} (a) = \sin (3 a)$

Step 1.2

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

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Step 1.2.1

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

$3 \sin (a) - 4 (\sin^{2} (a) \sin (a)) = \sin (3 a)$

Step 1.2.2

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

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Step 1.2.2.1

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

$3 \sin (a) - 4 (\sin^{2} (a) \sin^{1} (a)) = \sin (3 a)$

Step 1.2.2.2

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

$3 \sin (a) - 4 {\sin (a)}^{2 + 1} = \sin (3 a)$

Step 1.2.3

Add $2$ and $1$ .

$3 \sin (a) - 4 \sin^{3} (a) = \sin (3 a)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$a \approx 0, - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, - 0.3, 0.3$

Step 3