SOLUTION: If cos(t) = -2/3, and t is in quadrant III, find the exact values of sin(t), sec(t), csc(t), tan(t), and cot(t).

Algebra ->  Trigonometry-basics -> SOLUTION: If cos(t) = -2/3, and t is in quadrant III, find the exact values of sin(t), sec(t), csc(t), tan(t), and cot(t).       Log On


   

Question 1203393: If cos(t) = -2/3, and t is in quadrant III, find the exact values of
sin(t), sec(t), csc(t), tan(t), and cot(t).

Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
If +cos%28t%29+=+-2%2F3+, then +adj%2Fhyp=-2%2F3+

find opp
+%28-2%29%5E2+%2B+b%5E2+=+3%2A2+
+4+%2B+b%5E2+=+9+
+b%5E2+=+5+
+b+=+sqrt%285%29+

+sin%28t%29+=opp%2Fhyp+in quadrant III sin is negative
+sin%28t%29+=+-sqrt%285%29%2F3+

+tan%28t%29=sin%28t%29%2Fcos%28t%29=%28-sqrt%285%29%2F3%29%2F%28+-2%2F3%29=sqrt%285%29%2F2+



+sec%28t%29=1%2Fcos%28t%29=1%2F%28+-2%2F3%29=-3%2F2+






Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll use theta in place of t

cos(theta) = -2/3
cos(theta) = adjacent/hypotenuse

adjacent = -2
hypotenuse = 3

Use the pythagorean theorem a%5E2%2Bb%5E2=c%5E2 to determine the opposite side is sqrt%285%29.
I'll make this value negative to indicate we're below the x axis.

adjacent = -2
hypotenuse = 3
opposite = -sqrt%285%29

Here's what the diagram looks like


Then,
sin%28theta%29+=+opposite%2Fhypotenuse+=+-sqrt%285%29%2F3

tan%28theta%29+=+opposite%2Fadjacent+=+%28-sqrt%285%29%29%2F%28-2%29+=+sqrt%285%29%2F2

csc%28theta%29+=+hypotenuse%2Fopposite+=+3%2F%28-sqrt%285%29%29+=+-3%2Asqrt%285%29%2F5 (this is the reciprocal of sine)

sec%28theta%29+=+hypotenuse%2Fadjacent+=+%283%29%2F%28-2%29+=+-3%2F2 (this is the reciprocal of cosine)

(this is the reciprocal of tangent)

A similar question is found here:
https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1203332.html