SOLUTION: Find the exact value for cos(alpha+beta) if sin(alpha) = -3/5 and sin(beta) = 5/13 with alpha in quadrant 3 and beta in quadrant 1 Thank you smart people

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Question 1099718: Find the exact value for cos(alpha+beta) if sin(alpha) = -3/5 and sin(beta) = 5/13 with alpha in quadrant 3 and beta in quadrant 1
Thank you smart people
Found 3 solutions by Fombitz, MathTherapy, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28x%2By%29=cos%28x%29cos%28y%29-sin%28x%29sin%28y%29
and
cos%5E2%28x%29%2Bsin%5E2%28x%29=1
So then,
cos%5E2%28alpha%29%2Bsin%5E2%28alpha%29=1
cos%5E2%28alpha%29%2B9%2F25=1
cos%5E2%28alpha%29=16%2F25
Since alpha is in Q3, cosine is negative so,
cos%28alpha%29=-4%2F5
and similarly,
cos%5E2%28beta%29%2B25%2F169=1
cos%5E2%28beta%29=144%2F169
Since beta is in Q1, cosine is positive so,
cos%28beta%29=12%2F13
So putting it all together,
cos%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29
cos%28alpha%2Bbeta%29=%28-4%2F5%29%2812%2F13%29-%28-3%2F5%29%285%2F13%29
Work through that to get the answer.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Find the exact value for cos(alpha+beta) if sin(alpha) = -3/5 and sin(beta) = 5/13 with alpha in quadrant 3 and beta in quadrant 1
Thank you smart people




matrix%281%2C3%2C+sin+%28alpha%29%2C+%22=%22%2C+-+3%2F5%29, with alpha in QIII

As the above is a 3-4-5 PYTHAG. TRIPLE, we get: A = - 4, and so: matrix%281%2C3%2C+cos+%28alpha%29%2C+%22=%22%2C+-+4%2F5%29



matrix%281%2C3%2C+sin+%28beta%29%2C+%22=%22%2C+5%2F13%29, with β in QI

As the above is a 5-12-13 PYTHAG. TRIPLE, we get: A = 12, and so: matrix%281%2C3%2C+cos+%28beta%29%2C+%22=%22%2C+12%2F13%29

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
For many other similar solved problems on calculating trig functions see the lessons
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
    - Evaluating trigonometric expressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Trigonometry: Solved problems".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.