Trigonometry Examples

$\frac{1 - \cos (\frac{5 π}{6})}{\sin (\frac{5 π}{6})}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{1 - - \cos (\frac{π}{6})}{\sin (\frac{5 π}{6})}$

Step 1.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.3

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

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Step 1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

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

Step 1.4

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

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$2 + \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$