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# Calculus/Question

Question
Hello, thanks for taking the time to answer my question. It is:

Find the Exact value of sin(u+v) given that sin(u)=11/61 and cos(v)=-40/41 (Both us and v are in Quadrant II) and include the answer in Fractional form.

If you could please include as many of the steps you used to find your answer as possible would help. Thank you.

Hello again John,

Use the sum formula for sine, which is:
sin(u+v)=cos(u)sin(v)+cos(v)sin(u)

We know sin(u) and cos(v), now all we need is sin(v) and cos(u).
One way is to use the identity: [cos(x)]^2 + [sin(x)]^2 = 1

Thus, [cos(u)]^2 + [sin(u)]^2 = 1 ==> [cos(u)]^2 + [11/61)]^2 = 1
==> [cos(u)]^2 = 1-121/3721 = 3600/3721 ==> cos(u) = -sqrt(3600/3721)
==> cos(u) = -60/61, taking the negative square root since u is in Q2.

Similarly, [cos(v)]^2 + [sin(v)]^2 = 1 ==> [-40/41]^2 + [sin(v)]^2 = 1
==> [sin(v)]^2 = 1-1600/1681 = 81/1681 ==> sin(v) = sqrt(81/1681)
==> sin(v) = 9/41, taking the positive square root since v is also in Q2.

Now we can finish the problem.
sin(u+v) = (-60/61)*(9/41)+(-40/41)(11/61) = -980/2501

Good?

Abe

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Calculus

Volunteer

#### Abe Mantell

##### Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

##### Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook