Precalculus Examples

$2 \sin^{2} (60) + 3 \sin (60) + 1$

Step 1

Simplify each term.

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Step 1.1

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

$2 {(\frac{\sqrt{3}}{2})}^{2} + 3 \sin (60) + 1$

Step 1.2

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

$2 \frac{{\sqrt{3}}^{2}}{2^{2}} + 3 \sin (60) + 1$

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

$2 \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}} + 3 \sin (60) + 1$

Step 1.3.2

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

$2 \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}} + 3 \sin (60) + 1$

Step 1.3.3

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

$2 \frac{3^{\frac{2}{2}}}{2^{2}} + 3 \sin (60) + 1$

Step 1.3.4

Cancel the common factor of $2$ .

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Step 1.3.4.1

Cancel the common factor.

$2 \frac{3^{\frac{2}{2}}}{2^{2}} + 3 \sin (60) + 1$

Step 1.3.4.2

Rewrite the expression.

$2 \frac{3^{1}}{2^{2}} + 3 \sin (60) + 1$

Step 1.3.5

Evaluate the exponent.

$2 \frac{3}{2^{2}} + 3 \sin (60) + 1$

Step 1.4

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

$2 (\frac{3}{4}) + 3 \sin (60) + 1$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $4$ .

$2 \frac{3}{2 (2)} + 3 \sin (60) + 1$

Step 1.5.2

Cancel the common factor.

$2 \frac{3}{2 \cdot 2} + 3 \sin (60) + 1$

Step 1.5.3

Rewrite the expression.

$\frac{3}{2} + 3 \sin (60) + 1$

Step 1.6

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

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

Step 1.7

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

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

Step 2

Simplify the expression.

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Step 2.1

Write $1$ as a fraction with a common denominator.

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Add $3$ and $2$ .

$\frac{5}{2} + \frac{3 \sqrt{3}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2} + \frac{3 \sqrt{3}}{2}$

Decimal Form:

$5.09807621 \dots$