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

a.

33/65

b.

-33/65

c.

-63/65

d.

63/65

Answer 1

Use the following identity:

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

Given: cos(x) = -4/5

Placing cos(x) = -4/5 in the second quadrant, as stated in the question, we can use Pythagorean theorem to determine that the third side of the triangle is 3, and then we can determine the following:

sin(x) = 3/5

cos(x) = -4/5

tan(x) = 3/-4

Given: cos(t) = 12/13

Placing cos(x) = 12/13 in the fourth quadrant, as stated in the question, we can use Pythagorean theorem to determine that the third side of the triangle is -5, and then we can determine the following:

sin(x) = -5/13

cos(x) = 12/13

tan(x) = -5/12

Applying this information to the identity given above:

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

cos(x-t) = (-4/5)(12/13 + (3/5)(-5/13)

cos(x-t) = (-48/65) + (-15/65)

cos(x-t) = (-63/65)

So the answer should be -63/65.

A common mistake here would be not realizing that the third side of the second triangle is negative, not positive. This is because it forms the side corresponding to sin (parallel with the y-axis), which is negative in the fourth quadrant. If this is mistaken as a positive 5, the result is -33/65, which appears to be a choice as well. Watch out for that trap.

Hope this helps!

Answer 2

Need to find sin(x) and sin(t) first.

sin(x) = √(25-16) /5 = 3/5

sin(t) = √(169-144) /13 = 5/13

cos(x-t) = cos(x)cos(t)+sin(x)sin(t) by cosine angle subtraction formula

Just substitute in the values for the sines and cosines to get cos(x-t)