If sin(x) = 1/3 and sec(y) = 17/15 , where x and y lie between 0 and 𝜋 2 , evaluate the expression. cos(x − y)

Answer 1

cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y)

Triangle 1:

if sin(x) = 1/3, then draw a right triangle with hypotenuse 3 and the leg opposite angle x equal to 1. Then the leg adjacent to angle x is √8 = 2√2, by the Pythagorean Theorem.

Opposite side = 1

Adjacent side = 2√2

Hypotenuse = 3

Triangle 2:

If sec(y) = 17/15, then draw a right triangle with hypotenuse 17 and the leg adjacent to angle y equal to 15. Then the leg opposite angle y is 8 by the Pythagorean Theorem.

Opposite side = 8

Adjacent side = 15

Hypotenuse = 17

We find the following:

1) cos(x) = adjacent/hypotenuse of Triangle 1

= 2√2 / 3

2) cos(y) = 1/sec(y) = 15/17

3) sin(x) = 1/3 (given)

4) sin(y) = opposite/hypotenuse of Triangle 2

= 8/17

cos(x)*cos(y) + sin(x)*sin(y) = (2√2 / 3) * (15/17) + (1/3)*(8/17)

= (30√2 +8)/51