For positive acute angles A and B, it is know that tan A= 8/15 and sin B= 11/61. Find the value of cos (a+b) in simplest form

812/1037

Step-by-step explanation:

To solve this, we have to use trigonometric identities.

Cos (A + B) is given as Cos A Cos B - Sin A Sin B. And from the question, we have that

Tan A = 8/15.

We know that in a triangle, the Tan angle is represented Opp/Adj and thus the Opp is 8, and the Adj is 15. Using Pythagoras, we have

hyp² = opp² + adj²

hyp² = 8² + 15²

hyp² = 64 + 225

hyp² = 289

hyp = √289 = 17

The identity of Cos is Adj/Hyp and that of Sin is Opp/Hyp.

Cos A = 15/17

Sin A = 8/17

Repeating the same process for B, we have

Sin B = 11/61

adj² = hyp² - opp²

adj² = 61² - 11²

adj² = 3721 - 121

adj² = 3600

adj = √3600 = 60

Cos B = 60/61

Now, using the earlier stated trigonometric identity, we have

cos (a + b) = CosA CosB - SinA SinB

cos (a + b) = 15/17 * 60/61 - 8/17 * 11/61

cos (a + b) = 900/1037 - 88/1037

cos (a + b) = 812/1037

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