SOLUTION: Given ∠θ in standard position with its terminal arm in the stated quadrant, find the exact values of the remaining five trigonometric ratios for θ. cos θ = -2/3, quadrant II

Algebra ->  Trigonometry-basics -> SOLUTION: Given ∠θ in standard position with its terminal arm in the stated quadrant, find the exact values of the remaining five trigonometric ratios for θ. cos θ = -2/3, quadrant II      Log On


   

Question 1204873: Given ∠θ in standard position with its terminal arm in the stated quadrant, find the exact values of the remaining five trigonometric ratios for θ.
cos θ = -2/3, quadrant II
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

given:
cos%28theta%29=+-2%2F3, quadrant II
since cos%28theta%29=+adj%2Fhyp, we know that
adj=-2
hyp=3
then
opp=sqrt%283%5E2-%28-2%29%5E2%29
opp=sqrt%289-4%29
opp=sqrt%285%29

then

sin%28theta%29=opp%2Fhyp
sin%28theta%29=sqrt%285%29%2F3

In Quadrant II, ⁡ sin%28theta%29 is positive, so we need sin%28theta%29=sqrt%285%29%2F3

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

cot%28theta%29=cos%28theta%29%2Fsin%28theta%29
cot%28theta%29=%28-2%2F3%29%2F%28sqrt%285%29%2F3%29
cot%28theta%29=-2%2F%28sqrt%285%29%29
cot%28theta%29=-%282sqrt%285%29%29%2F5


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


csc%28theta%29=1%2Fsin%28theta%29
csc%28theta%29=1%2F%28sqrt%285%29%2F3%29
csc%28theta%29=3%2Fsqrt%285%29
csc%28theta%29=3sqrt%285%29%2F5