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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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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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Mathematics
3 years
2021-08-28T11:16:35+00:00
2021-08-28T11:16:35+00:00 1 Answers
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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