Expand Using Sum/Difference Formulas cos((4pi)/3)

$\cos (\frac{4 π}{3})$

The angle $\frac{4 π}{3}$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\cos (\frac{4 π}{3} + 0)$

Use the sum formula for cosine to simplify the expression. The formula states that $\cos (A + B) = - (\cos (A) \cos (B) + \sin (A) \sin (B))$ .

$\cos (0) \cdot \cos (\frac{4 π}{3}) - \sin (0) \cdot \sin (\frac{4 π}{3})$

The exact value of $\cos (0)$ is $1$ .

$(1) \cdot \cos (\frac{4 π}{3}) - \sin (0) \cdot \sin (\frac{4 π}{3})$

Simplify $\cos (\frac{4 π}{3})$ .

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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.

$(1) \cdot (- \cos (\frac{π}{3})) - \sin (0) \cdot \sin (\frac{4 π}{3})$

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

$(1) \cdot (- \frac{1}{2}) - \sin (0) \cdot \sin (\frac{4 π}{3})$

The exact value of $\sin (0)$ is $0$ .

$(1) \cdot (- \frac{1}{2}) + 0 \cdot \sin (\frac{4 π}{3})$

Simplify $\sin (\frac{4 π}{3})$ .

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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 third quadrant.

$(1) \cdot (- \frac{1}{2}) + 0 \cdot (- \sin (\frac{π}{3}))$

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

$(1) \cdot (- \frac{1}{2}) + 0 \cdot (- \frac{\sqrt{3}}{2})$

Simplify each term.

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Multiply $- \frac{1}{2}$ by $1$ .

$- \frac{1}{2} + 0 \cdot (- \frac{\sqrt{3}}{2})$

Multiply $0 (- \frac{\sqrt{3}}{2})$ .

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Multiply $- 1$ by $0$ .

$- \frac{1}{2} + 0 \frac{\sqrt{3}}{2}$

Multiply $0$ by $\frac{\sqrt{3}}{2}$ .

$- \frac{1}{2} + 0$

Add $- \frac{1}{2}$ and $0$ .

$- \frac{1}{2}$

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