Question

The angle θ is in the first quadrant of the unit circle and tan(θ)= $\frac{8}{15}$ .Since tan(θ)=$\frac{sin(θ)}{cos(θ)}$ , does this mean that sin(θ)=8 and cos(θ)=15? If not, clearly explain and give the exact values of sin(θ) and cos(θ).

Answer 1

Answer:

$\sin(\theta)=0.47$ and $\cos(\theta)=0.88$.

Step-by-step explanation:

Given:

Angle lies in first quadrant.

$\tan\theta=\frac{8}{15}$

Since, $\tan\theta=\frac{\cos(\theta}{\sin(\theta)}$ this does not mean that $\sin(\theta)=8$ and  $\cos(\theta)=15$, because it is the ratio of the $\sin(\theta)$ and $\cos(\theta)$ which is not necessarily the exact values of them. And also $\sin(\theta)$ and $\cos(\theta)$ values lie between $-1 \ to\ 1$ and so it cannot be $=8\ or\ 15$.

In order to find the exact values, we need to find the exact angle.

$\tan\theta=\frac{8}{15}$

So,

$\theta=\tan^{-1}(\frac{8}{15})$

$\theta= 28.07\°$   As the angle lies in 1st quadrant.

Using the angle $\theta$ we can find the exact values for $\sin(\theta)$ and $\cos(\theta)$.

$sin(28.07\°)=0.47$

$cos(28.07\°)=0.88$