Question 1077470: Hi can someone help me with this problem please? THANK YOU!
a and B are acute angles with tan(a)=8/15 and sec(B)=41/40.
Find cos(a+B).
Select One:
a. 185/697
b. 672/697
c. 528/697
d. 455/697
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! To calculate  you could use the trigonometric identity
 .
If  and  are acute angles they could be part of a right triangle,
and/or because they are acute angles, you know their sines and cosines are positive.
A right triangle with leg lengths  and  ,
and angle opposite the leg of length
would have  .
Its hypotenuse length would be  .
You can find that using the Pytahgorean theorem,
or from a list of Pythagorean triples.
(I searched for that list online, because I could.
Had I been in class with just a calculator,
I would not have had that easier option).
So, it would have  and  .
You could also use trigonometric identities
to calculate sine and cosine based on the value of tangent,
but those trigonometric identities are not the ones people usually remember.
As for B,  , so  .
You can use the Pytahgorean theorem, or the popular trigonometric identity 
to find that  .
(I searched in the list of Pythagorean triples for the (9,40,41) triple,
which tells you that a right triangle with hypotenuse=41,
and one leg=40, has other leg=9).
Now that we know both sines and cosines, substituting into
, we get
 .
At the point where I realized that the numerator was going to be even, not odd,
I could have eliminated options a and d.
If pressed for time in a test, that would be helpful.
To tell between b and c, I had to do some calculations.
(Just in case I could have made an arithmetic mistake,
I used the computer as a fancy calculator
to find the angles that had and ;
add them; find the cosine of the sum;
express that number as a fraction with a 3-digit denominator,
and multiply the number times the denominator to see if it was exactly that fraction).
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