Question

For positive acute angles

Answer 1

9514 1404 393

Answer:

  77/85

Step-by-step 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

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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

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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

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Some graphing calculators can express the result ...

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

  ... as the fraction 77/85