precalc questions

Question

Find the values of all the trigonometric functions from the given information.
csc t = −4/3, 3 pi /2 < t < 2π
sin t =

cos t =


tan t =


cot t =


sec t =

Answer 1

Based on the interval you have given, we are in the 4th quadrant.  The only trig function that positive in this quadrant is cosine.

 

 

csc t = -4/3

 

1 / sin t = -4 / 3

 

sin t = -3 / 4

 

 

To get the other trig functions , we use the trigonometric identities:

 

sin²t + cos²t = 1               and             tant =  sint / cost

 

 

cos t = √(1 - sin²t)

 

cos t = √(1 - (-3 / 4)²)

 

cos t = √(1 - (9 / 16))

 

cos t = √((16 / 16) - (9 / 16))

 

cos t = √(7 / 16)

 

cos t = √(7) / 4

 

 

tan t = (-3 / 4) / (√(7) / 4)

 

tan t = (-3 / 4) * (4 / √(7))

 

tan t = -3 / √7

 

 

Rationalize the denominator to get

 

tan t = (-3√(7)) / 7

 

 

To find cot t, take the inverse of tan t that we just found.

To find sec t, evaluate 1 / cos t.