Find cos (x - t) given that the cos (x) =-4/5, with x in quadrant 2, and cos (t) = 12/13 , with t in quadrant 4

Question

Find cos (x - t) given that the cos (x) =-4/5, with x in quadrant 2, and cos (t) = 12/13 , with t in quadrant 4

Answers: 

a. -33/65
b. -63/65
c. 33/65
d. 63/65

Answer 1

Use the sum and difference of two angles identity:

 

cos(x-t) = cos(x)cos(t) + sin(x)sin(t)

cos(x-t) = (-4/5)(12/13) + sin(x)sin(t)

cos(x-t) = -48/65 + sin(x)sin(t)       ← Equation (1)

 

To find sin(x) and sin(t) use the Pythagorean Identity:

 

sin²(x) + cos²(x) = 1

sin²(x) = 1 - cos²(x)

sin²(x) = 1 - (-4/5)²

sin²(x) = 1 - 16/25

sin²(x) = 9/25

sin(x) = ±3/5

In Quadrant 2, the sine is positive, so sin(x) = 3/5

 

Doing the same for sin(t):

sin²(t) = 1 - (12/13)²

sin(t) = ±5/13

In Q IV, the sine is negative, so sin(t) = -5/13

 

So sin(x)sin(t) = (3/5)(-5/13) = -15/65.  Plug it into equation 1 above and compute the answer.