Assume that theta is a positive acute angle. Given: sin theta = 4/5. Find: cos 2theta

1. Identify the problem type: This is a trigonometry problem involving a double angle identity.
2. Recall the double angle identity for cosine: There are three common forms for the double angle identity for cosine:
- $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
- $\cos(2\theta) = 2\cos^2(\theta) - 1$
- $\cos(2\theta) = 1 - 2\sin^2(\theta)$
3. Choose the most suitable identity: Since we are given the value of $\sin(\theta)$, the identity $\cos(2\theta) = 1 - 2\sin^2(\theta)$ is the most direct to use.
4. Substitute the given value of $\sin(\theta)$ into the identity: We are given that $\sin(\theta) = \frac{4}{5}$. Substituting this into the chosen identity:
$$\cos(2\theta) = 1 - 2\left(\frac{4}{5}\right)^2$$
5. Calculate the square of $\sin(\theta)$:
$$\left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2} = \frac{16}{25}$$
6. Substitute the squared value back into the equation:
$$\cos(2\theta) = 1 - 2\left(\frac{16}{25}\right)$$
7. Perform the multiplication:
$$\cos(2\theta) = 1 - \frac{32}{25}$$
8. Perform the subtraction: To subtract the fractions, find a common denominator, which is 25.
$$\cos(2\theta) = \frac{25}{25} - \frac{32}{25}$$
$$\cos(2\theta) = \frac{25 - 32}{25}$$
$$\cos(2\theta) = \frac{-7}{25}$$
9. Consider the condition that $\theta$ is a positive acute angle: Since $\theta$ is a positive acute angle, it lies in the first quadrant ($0 < \theta < \frac{\pi}{2}$). Therefore, $2\theta$ will be in the range $0 < 2\theta < \pi$. In this range, $\cos(2\theta)$ can be positive or negative. Our calculated value of $\frac{-7}{25}$ is a valid result for $\cos(2\theta)$.