Given:
[tex]\csc (\theta)=\frac{5}{3}\text{ and }\cos (\theta)<0[/tex]
We want to find the value of tan(θ).
First, remember that cos(θ) is negative in the quadrants II and III of the xy-plane. Notice that as cscθ is positive (5/3), our reference angle should be located at the quadrant II.
We have to also remember that csc(θ) is a function that relations the hypotenuse of a right triangle and the opposite side to θ.
So, we could draw this situation here below:
Now, remember that tan(θ) is defined as the ratio between the opposite side to θ and its adjacent side.
So, we might need to find the adjacent side to θ in the triangle given. For this, we could use the Pythagorean theorem as follows:
As you can see, the adjacent side (a), is 4.
Then,
[tex]\tan \theta=\frac{3}{4}[/tex]
And, the value of θ will be:
[tex]\theta=\arctan (\frac{3}{4})=36.86[/tex]
