Find the exact values of each of the remaining trigonometric functions of theta.

Question

sec(theta)=2, tan(theta)>0.

 

Answers:

 

1. sin(theta)=?

2.cos(theta)=?

3.tan(theta)=?

4.csc(theta)=?

5.cot(theta)=?

Answer 1

secθ=2 (or 2/1) (positive) & tanθ>0 (positive), this tells you that the angle/triangle will be drawn in the first quadrant where secθ and tanθ are both positive.

 

Note that secθ=hypotenuse/adjacent side=2/1, so 2=the hypotenuse & 1=the adjacent side...now use a²+b²=c²

(opp)² + (adj)²= (hyp)²

a²+(1)²=(2)²

a²+1=4 (subtract the one from both sides)

a²=3 (square root both sides)

a=√3 (this is the side opposite of θ)

 

a=√3 (opp), b=1 (adj), c=2 (hyp).  Now just fill in the ratios!!!

 

Answers: sinθ=a/c = √3/2  

              cosθ=b/c = 1/2  

              tanθ=a/b=√3/1 = √3 

              cscθ=c/a=2/√3 (multiply top and bottom by √3 to get radical out of the denominator) = 2√3/3

              secθ=c/b=2/1 = 2

              cotθ=b/a=1/√3(multiply top and bottom by √3 to get radical out of the denominator) = √3/3

 

I hope this helps!!