Trigonometry Examples

$8 (\cos (\frac{5 π}{3}) + \sin (\frac{5 π}{3}))$

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.

$8 (\cos (\frac{π}{3}) + \sin (\frac{5 π}{3}))$

Step 1.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$8 (\frac{1}{2} + \sin (\frac{5 π}{3}))$

Step 1.3

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.

$8 (\frac{1}{2} - \sin (\frac{π}{3}))$

Step 1.4

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

$8 (\frac{1}{2} - \frac{\sqrt{3}}{2})$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$8 (\frac{1}{2}) + 8 (- \frac{\sqrt{3}}{2})$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $8$ .

$2 (4) \frac{1}{2} + 8 (- \frac{\sqrt{3}}{2})$

Step 2.2.2

Cancel the common factor.

$2 \cdot 4 \frac{1}{2} + 8 (- \frac{\sqrt{3}}{2})$

Step 2.2.3

Rewrite the expression.

$4 + 8 (- \frac{\sqrt{3}}{2})$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$4 + 8 \frac{- \sqrt{3}}{2}$

Step 2.3.2

Factor $2$ out of $8$ .

$4 + 2 (4) \frac{- \sqrt{3}}{2}$

Step 2.3.3

Cancel the common factor.

$4 + 2 \cdot 4 \frac{- \sqrt{3}}{2}$

Step 2.3.4

Rewrite the expression.

$4 + 4 (- \sqrt{3})$

Step 2.4

Multiply $- 1$ by $4$ .

$4 - 4 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$4 - 4 \sqrt{3}$

Decimal Form:

$- 2.92820323 \dots$