Question

What is the value of sin © and tan o if cos © Зл and << 27? 2 5

Answer 1

Let's see where the angle is first:

Angle theta will be in the yellow region. The angle in red is an example. For that angle, it's cosine is the proyection of the line that forms the angle over the x-axis and it's sine is the proyection over the y-axis. I'll add them in blue and green.

So the cosine is positive, and the sine is negative for that angle.

We can find the angle by using the inverse of the cosine:

$\begin{gathered} \cos \theta=\frac{4}{5} \\ \cos ^{-1}(\frac{4}{5})=\theta \end{gathered}$

But we have to be careful with this because the result will not be in the quadrant we want. The results of these operations are always in the first quadrant (between 0 and pi/2)

This one is:

$\theta\approx0.6435\text{ rad}$

To find the angle that is beween 3pi/2 and 2pi, we have to substract that angle from 2pi:

$2\pi-0.6435\approx5.6397rad$

And with that angle we can calculate the sine:

$\sin (\theta)=-\frac{3}{5}$

(I solved it with a calculator so I don't miss decimals and get a more exact answer)

The tangent is the quotient between the sine and the cosine:

$\tan (\theta)=\frac{\sin\theta}{\cos\theta}=\frac{\frac{4}{5}}{-\frac{3}{5}}=-\frac{4\times5}{3\times5}=-\frac{4}{3}$