In the problem, t is a real number and P = (x, y) is the point on the unit circle that corresponds to t. Find the exact value of the indicated trigonometric

Question

In the problem, t is a real number and P = (x, y) is the
point on the unit circle that corresponds to t. Find the exact
value of the indicated trigonometric function of t.

 (-√65/9, -4/9)  Find cot t.

(4/7, -√33/7) Find csc t.

 

(5/8, √39/8) Find tan t.

Answer 1

Hi Couva,

 

The question is basically asking us to find the trigonometric functions of the angle that defines point t given in different quadrants. 

 

I) (-√65/9, -4/9) This is in quadrant III. I will use tangent function to find the angle theta.

If the angle is theta, then (pi+theta) = tan(theta).

tan(theta) = (-4/9) / (-√65/9)

= 4 / √65

cot theta = 1/tan t = √65/4

 

II) (4/7, -√33/7) Find csc t.

This is in quadrant IV. csc t = 1 / sin t

Again, if the angle is theta, sin theta = -(4/7) / 1  (hypotenuse is 1)

csc theta = 1/-(4/7) = -7/4 

 

III) (5/8, √39/8) Find tan t.

This is in quadrant I. Everything is positive here.

tan theta = (√39/8) / (5/8)

= √39/5