If $$ \theta=\frac{4\pi}{3} $$, find the exact value of each expression below.

(a) cos2θ=

(b) 2cosθ=

(c) cosθ2=




Given $\theta = \frac{4\pi}{3}$, we need to find the exact values for the following expressions:

### (a) $\cos 2\theta$
First, calculate $2\theta$:
\[ 2\theta = 2 \times \frac{4\pi}{3} = \frac{8\pi}{3} \]

Since the cosine function is periodic with a period of $2\pi$, we can reduce $\frac{8\pi}{3}$ to an equivalent angle within $[0, 2\pi)$ by subtracting $2\pi$:
\[ \frac{8\pi}{3} - 2\pi = \frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3} \]

Now, we find $\cos \frac{2\pi}{3}$. Since $\frac{2\pi}{3}$ is in the second quadrant where cosine is negative, and $\cos \frac{\pi}{3} = \frac{1}{2}$:
\[ \cos \frac{2\pi}{3} = -\cos \frac{\pi}{3} = -\frac{1}{2} \]

Therefore:
\[ \cos 2\theta = -\frac{1}{2} \]

### (b) $2\cos \theta$
First, find $\cos \theta$ for $\theta = \frac{4\pi}{3}$. Since $\frac{4\pi}{3}$ is in the third quadrant where cosine is negative, and $\cos \frac{\pi}{3} = \frac{1}{2}$:
\[ \cos \frac{4\pi}{3} = -\cos \frac{\pi}{3} = -\frac{1}{2} \]

Now multiply by 2:
\[ 2\cos \theta = 2 \times -\frac{1}{2} = -1 \]

### (c) $\cos \frac{\theta}{2}$
First, calculate $\frac{\theta}{2}$:
\[ \frac{\theta}{2} = \frac{1}{2} \times \frac{4\pi}{3} = \frac{2\pi}{3} \]

Now, we find $\cos \frac{2\pi}{3}$ as calculated before:
\[ \cos \frac{2\pi}{3} = -\frac{1}{2} \]

Therefore:
\[ \cos \frac{\theta}{2} = -\frac{1}{2} \]

To summarize:
(a) $\cos 2\theta = -\frac{1}{2}$

(b) $2\cos \theta = -1$

(c) $\cos \frac{\theta}{2} = -\frac{1}{2}$

Would you like details or have any questions? Here are some additional questions you might be interested in:

1. How do you find the sine of an angle given in radians?
2. What is the general formula for the double angle identities?
3. How can you derive the cosine of an angle using the unit circle?
4. What are the properties of trigonometric functions in different quadrants?
5. How do you convert between degrees and radians?

**Tip:** Remember that trigonometric functions are periodic, which means they repeat their values in regular intervals. This can be useful for simplifying calculations!

Find the exact values of cosine double angle and half angle identities for theta equals 4pi over 3. Includes detailed step-by-step solutions and explanations for cos 2theta, 2cos theta, and cos theta/2.

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