Precalculus Examples

$\sin (\frac{3 π}{2}) \tan (- \frac{8 π}{3}) + \cos (- \frac{5 π}{6})$

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. Make the expression negative because sine is negative in the fourth quadrant.

$- \sin (\frac{π}{2}) \tan (- \frac{8 π}{3}) + \cos (- \frac{5 π}{6})$

Step 1.2

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

$- 1 \cdot 1 \tan (- \frac{8 π}{3}) + \cos (- \frac{5 π}{6})$

Step 1.3

Multiply $- 1$ by $1$ .

$- 1 \tan (- \frac{8 π}{3}) + \cos (- \frac{5 π}{6})$

Step 1.4

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- 1 \tan (\frac{4 π}{3}) + \cos (- \frac{5 π}{6})$

Step 1.5

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

$- 1 \tan (\frac{π}{3}) + \cos (- \frac{5 π}{6})$

Step 1.6

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

$- 1 \sqrt{3} + \cos (- \frac{5 π}{6})$

Step 1.7

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$- \sqrt{3} + \cos (- \frac{5 π}{6})$

Step 1.8

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \sqrt{3} + \cos (\frac{7 π}{6})$

Step 1.9

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 third quadrant.

$- \sqrt{3} - \cos (\frac{π}{6})$

Step 1.10

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

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

Step 2

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

Combine fractions.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

Subtract $\sqrt{3}$ from $- 2 \sqrt{3}$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 2.59807621 \dots$