Question

For positive acute angles

A
A and
B
,
B, it is known that
cos
⁡
A
=
4
5
cosA=
5
4
​
and
sin
⁡
B
=
15
17
.
sinB=
17
15
​
. Find the value of
cos
⁡
(
A
−
B
)
cos(A−B) in simplest form.

Answer 1

9514 1404 393

Answer:

77/85

Explanation:

Given:

cos(A) = 4/5

sin(B) = 15/17

Find:

cos(A-B)

Solution:

From your knowledge of Pythagorean triples, you know that one of them is (3, 4, 5) and another is (8, 15, 17). Using your imagination, or by drawing the triangles, you can determine the needed trig functions to be ...

sin(A) = 3/5

cos(B) = 8/17

__

Alternatively, you can use the Pythagorean identity to find ...

sin(A) = √(1 -cos²(A)) = √(1 -(4/5)²) = √(9/25) = 3/5

cos(B) = √(1 -sin²(B)) = √(1 -(15/17)²) = √(64/289) = 8/17

__

Then application of the angle difference formula is straightforward.

cos(A -B) = cos(A)cos(B) +sin(A)sin(B)

cos(A -B) = (4/5)(8/17) +(3/5)(15/17) = (32 +45)/85

cos(A -B) = 77/85

_____

Some graphing calculators can express the result ...

cos(arccos(4/5) -arcsin(15/17)) ≈ 0.90588235294

... as the fraction 77/85