Question

23. Find the exact value of cos(20) knowing that: *

Answer 1

Answer:

$\cos(2\theta) = 0$

Explanation:

Given

$\sin(\theta) = -\frac{\sqrt 2}{2}$

$270^o < \theta < 360^o$

Required

$\cos(2\theta)$

Recall that:

$\sin^2(\theta) + \cos^2(\theta) =1$

Substitute: $\sin(\theta) = -\frac{\sqrt 2}{2}$

$(-\frac{\sqrt 2}{2})^2 + \cos^2(\theta) =1$

$\frac{1}{2} + \cos^2(\theta) =1$

Collect like terms

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

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

$\cos(2\theta)$ is calculated as:

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

So, we have:

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

$\cos(2\theta) = \frac{1}{2} - \frac{2}{4}$

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

$\cos(2\theta) = 0$