Question

Let theta be an angle in quadrant IV such that sin theta= -5/13 Find the exact values of sec theta and tan theta

Answer 1

13^2-5^2=12^2
Quad 4: sec is positive, tan is negative
sec=1/cos
cos=12/13
sec=13/12
tan=sin/cos
tan=-5/12

Answer 2

`sec\theta =(13)/(12)\\\\tan\theta =-(5)/(12)`

Explanation:

We have value of sin θ = -5/13 and angle in quadrant IV .

In quadrant IV cos θ and sec θ are positive all others are negative.

We have

`sin\theta =-(5)/(13)\\\\sin^2\theta+cos^2\theta=1\\\\\left ( -(5)/(13)\right )^2+cos^2\theta=1\\\\cos^2\theta =(144)/(169)\\\\cos\theta=(12)/(13)\\\\sec\theta =(1)/(cos\theta )=(13)/(12)\\\\tan\theta =(sin\theta)/(cos\theta)=((-5)/(13))/((12)/(13))=-(5)/(12)`

`sec\theta =(13)/(12)\\\\tan\theta =-(5)/(12)`