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

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2 Answers
Feb 14, 2018

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.

Feb 14, 2018

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