Trigonometry Examples

$\sin^{2} (150) + \cos^{2} (30)$

Step 1

Simplify each term.

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Step 1.1

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

$\sin^{2} (30) + \cos^{2} (30)$

Step 1.2

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

${(\frac{1}{2})}^{2} + \cos^{2} (30)$

Step 1.3

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{2}}{2^{2}} + \cos^{2} (30)$

Step 1.4

One to any power is one.

$\frac{1}{2^{2}} + \cos^{2} (30)$

Step 1.5

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

$\frac{1}{4} + \cos^{2} (30)$

Step 1.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{4} + {(\frac{\sqrt{3}}{2})}^{2}$

Step 1.7

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

$\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}$

Step 1.8

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

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Step 1.8.1

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

$\frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 1.8.2

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

$\frac{1}{4} + \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 1.8.3

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

$\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}$

Step 1.8.4

Cancel the common factor of $2$ .

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Step 1.8.4.1

Cancel the common factor.

$\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}$

Step 1.8.4.2

Rewrite the expression.

$\frac{1}{4} + \frac{3^{1}}{2^{2}}$

Step 1.8.5

Evaluate the exponent.

$\frac{1}{4} + \frac{3}{2^{2}}$

Step 1.9

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

$\frac{1}{4} + \frac{3}{4}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 + 3}{4}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $1$ and $3$ .

$\frac{4}{4}$

Step 2.2.2

Divide $4$ by $4$ .

$1$