Question

q7 (2 points)
determine the exact value of $cos(\theta)$ given that $\tan(\theta) = 4$ and $pi < \theta < \frac{3\pi}{2}$.

Explanation:

Step1: Use tan to sin/cos identity

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = 4 \implies \sin(\theta) = 4\cos(\theta)$

Step2: Apply Pythagorean identity

Substitute $\sin(\theta)=4\cos(\theta)$ into $\sin^2(\theta) + \cos^2(\theta) = 1$:
$(4\cos(\theta))^2 + \cos^2(\theta) = 1$
$16\cos^2(\theta) + \cos^2(\theta) = 1$
$17\cos^2(\theta) = 1$
$\cos^2(\theta) = \frac{1}{17}$
$\cos(\theta) = \pm\frac{1}{\sqrt{17}} = \pm\frac{\sqrt{17}}{17}$

Step3: Determine sign of cosine

$\pi < \theta < \frac{3\pi}{2}$ (third quadrant), where cosine is negative:
$\cos(\theta) = -\frac{\sqrt{17}}{17}$

Answer:

$-\frac{\sqrt{17}}{17}$