Precalculus Examples

$3 \csc (\frac{2 π}{3}) + \sin (\frac{5 π}{6})$

Step 1

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

$3 \csc (\frac{π}{3}) + \sin (\frac{5 π}{6})$

Step 2

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

$3 \frac{2}{\sqrt{3}} + \sin (\frac{5 π}{6})$

Step 3

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

$3 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) + \sin (\frac{5 π}{6})$

Step 4

Combine and simplify the denominator.

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Step 4.1

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

$3 \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} + \sin (\frac{5 π}{6})$

Step 4.2

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

$3 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + \sin (\frac{5 π}{6})$

Step 4.3

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

$3 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + \sin (\frac{5 π}{6})$

Step 4.4

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

$3 \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + \sin (\frac{5 π}{6})$

Step 4.5

Add $1$ and $1$ .

$3 \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} + \sin (\frac{5 π}{6})$

Step 4.6

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

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Step 4.6.1

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

$3 \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + \sin (\frac{5 π}{6})$

Step 4.6.2

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

$3 \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \sin (\frac{5 π}{6})$

Step 4.6.3

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

$3 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + \sin (\frac{5 π}{6})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$3 \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + \sin (\frac{5 π}{6})$

Step 4.6.4.2

Rewrite the expression.

$3 \frac{2 \sqrt{3}}{3^{1}} + \sin (\frac{5 π}{6})$

Step 4.6.5

Evaluate the exponent.

$3 \frac{2 \sqrt{3}}{3} + \sin (\frac{5 π}{6})$

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Cancel the common factor.

$3 \frac{2 \sqrt{3}}{3} + \sin (\frac{5 π}{6})$

Step 5.2

Rewrite the expression.

$2 \sqrt{3} + \sin (\frac{5 π}{6})$

Step 6

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

$2 \sqrt{3} + \sin (\frac{π}{6})$

Step 7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$3.96410161 \dots$