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

Question

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

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Kiệt Gia 3 years 2021-08-28T11:16:35+00:00 1 Answers 29 views 0

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    2021-08-28T11:17:47+00:00

    Answer:

    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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