Find the Exact Value sin(120 degrees )^2

$\sin^{2} (120 °)$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sin^{2} (60)$

Step 2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

${(\frac{\sqrt{3}}{2})}^{2}$

Step 3

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{3^{\frac{2}{2}}}{2^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2^{2}}$

Rewrite the expression.

$\frac{3^{1}}{2^{2}}$

Evaluate the exponent.

$\frac{3}{2^{2}}$

Step 5

Raise $2$ to the power of $2$ .

$\frac{3}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{4}$

Decimal Form:

$0.75$