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For positive acute angles A and B, it is known that cos A = 12/13 and sin B = 21/29Find the value of cos(A - B) in simplest form.

User Utdemir
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1 Answer

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To solve this problem we must apply the cosine formula for sum of angles, which is:

cos(A ± B) = cos A . cos B ± sin A . sin B

For this case we may write:

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

Now let's plug the given information in the equation above. We get:

cos(A - B) = 12/13 . cos B + sin A . 21/29 (*)

Now we must find the unknown values. Apply fundamental trigonometric relation:

sin ² A + cos ² A = 1 (I)

and

sin ² B + cos ² B = 1 (II)

If cos A = 12/13 then equation (I) becomes:

sin² A + (12/13)² = 1

sin² A = 1 - (12/13)²

sin² A = 1 - 144/169

sin² A = 169/169 - 144/169

sin² A = 25/169

Since angle A is a positive angle, we get:

sin A = 5/13

Now, let's do the same thing for equation (II). If sin B = 21/29 , we have:

(21/29)² + cos² B = 1

cos² B = 1 - (21/29)²

cos² B = 1 - 441/841

cos² B = 841/841 - 441/841

cos² B = 400/841

Since B is positive angle, we get:

cos B = 20/29

Now we can go back the original equation (*). We get:

cos(A - B) = 12/13 . cos B + sin A . 21/29

cos(A - B) = 12/13 . 20/29 + 5/13 . 21/29

cos(A - B) = 240/377 + 105/377

cos(A - B) = 345/377

Answer: cos(A - B) = 345/377

User Arinzehills
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