Precalculus Examples

$\sqrt{3} \cos (30) + 1 \sin (30)$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{3} \frac{\sqrt{3}}{2} + 1 \sin (30)$

Step 1.2

Multiply $\sqrt{3} \frac{\sqrt{3}}{2}$ .

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Step 1.2.1

Combine $\sqrt{3}$ and $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3} \sqrt{3}}{2} + 1 \sin (30)$

Step 1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} \sqrt{3}}{2} + 1 \sin (30)$

Step 1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2} + 1 \sin (30)$

Step 1.2.4

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

$\frac{{\sqrt{3}}^{1 + 1}}{2} + 1 \sin (30)$

Step 1.2.5

Add $1$ and $1$ .

$\frac{{\sqrt{3}}^{2}}{2} + 1 \sin (30)$

Step 1.3

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

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Step 1.3.1

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

$\frac{{(3^{\frac{1}{2}})}^{2}}{2} + 1 \sin (30)$

Step 1.3.2

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

$\frac{3^{\frac{1}{2} \cdot 2}}{2} + 1 \sin (30)$

Step 1.3.3

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

$\frac{3^{\frac{2}{2}}}{2} + 1 \sin (30)$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$\frac{3^{\frac{2}{2}}}{2} + 1 \sin (30)$

Step 1.3.4.2

Rewrite the expression.

$\frac{3^{1}}{2} + 1 \sin (30)$

Step 1.3.5

Evaluate the exponent.

$\frac{3}{2} + 1 \sin (30)$

Step 1.4

Multiply $\sin (30)$ by $1$ .

$\frac{3}{2} + \sin (30)$

Step 1.5

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

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

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $3$ and $1$ .

$\frac{4}{2}$

Step 2.2.2

Divide $4$ by $2$ .

$2$