Question

Find the exact value of `tan(\theta)` if `csc(\theta)=-4/3` and the terminal side of the `\theta` lies in the third quadrant?

Answer 1

You marked the correct answer; `tan(theta)=(3sqrt7)/7`

Explanation:

Using the reciprocal identity

`cscx=1/sinx`,

we can rewrite the given statement:

`csc(theta)=-4/3`

`=>1/sin(theta)=-4/3`

`=>sintheta=-3/4`

This means that the ratio of the opposite side to the hypotenuse is `-3:4`. Knowing this and using the Pythagorean theorem, we can find the relative ratio of the adjacent side:

`a^2+b^2=c^2`

`=>(-3)^2+b^2=4^2`

`9+b^2=16`

`b^2=7`

`b=+-sqrt7`

Since this is in the third quadrant, this value must be negative, so

`b=-sqrt7`

Now, since we know that `tan` is the ratio between the opposite to the adjacent, we can write that `tan(theta)=3/sqrt7`. After rationalizing the denominator, this becomes `(3sqrt7)/7`.

Answer 2

`tantheta=(3sqrt7)/7`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)sintheta=1/csctheta" and "tantheta=sintheta/costheta`

`•color(white)(x)sin^2theta+cos^2theta=1`

`rArrcostheta=+-sqrt(1-sin^2theta)`

`csctheta=-4/3rArrsintheta=-3/4`

`theta" is in third quadrant where "costheta<0`

`rArrcostheta=-sqrt(1-(-3/4)^2)`

`color(white)(rArrcostheta)=-sqrt(7/16)=-sqrt7/4`

`rArrtantheta=-3/4xx-4/sqrt7=3/sqrt7=(3sqrt7)/7`