SOLUTION: I am studying for ACTs and am having trouble with this question. I am told to give the answer in Fractional Form. Find the Exact value of sin(u + v) given that sin(u) = 11/61 an

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Question 828893: I am studying for ACTs and am having trouble with this question. I am told to give the answer in Fractional Form.
Find the Exact value of sin(u + v) given that sin(u) = 11/61 and cos(v) = -40/41 (and that both u and v are in Quadrant II).
I tried messing around with the problem a bit, but I couldn't get anything to really work. Help is appreciated (please don't just put the answer, include the steps on how you got that fraction).
Answer by jim_thompson5910(35256) (Show Source):
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I am studying for ACTs and am having trouble with this question. I am told to give the answer in Fractional Form.
Find the Exact value of sin(u + v) given that sin(u) = 11/61 and cos(v) = -40/41 (and that both u and v are in Quadrant II).
I tried messing around with the problem a bit, but I couldn't get anything to really work. Help is appreciated (please don't just put the answer, include the steps on how you got that fraction).

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"both u and v are in Quadrant II", so...
sin(u) > 0, sin(v) > 0
cos(u) < 0, cos(v) < 0

We'll start off using the identity sin^2(x) + cos^2(x) = 1 to find cos(u)

sin^2(x) + cos^2(x) = 1

sin^2(u) + cos^2(u) = 1

(sin(u))^2 + (cos(u))^2 = 1

(11/61)^2 + (cos(u))^2 = 1

121/3721 + (cos(u))^2 = 1

(cos(u))^2 = 1 - 121/3721

(cos(u))^2 = 3721/3721 - 121/3721

(cos(u))^2 = (3721 - 121)/3721

(cos(u))^2 = 3600/3721

cos(u) = -sqrt(3600/3721) ... note: cos(u) < 0 since we're in quadrant II

cos(u) = -sqrt(3600)/sqrt(3721)

cos(u) = -60/61

Then we'll use the identity sin^2(x) + cos^2(x) = 1 to find sin(v)

sin^2(x) + cos^2(x) = 1

sin^2(v) + cos^2(v) = 1

(sin(v))^2 + (cos(v))^2 = 1

(sin(v))^2 + (-40/41)^2 = 1

(sin(v))^2 + 1600/1681 = 1

(sin(v))^2 = 1 - 1600/1681

(sin(v))^2 = 1681/1681 - 1600/1681

(sin(v))^2 = (1681 - 1600)/1681

(sin(v))^2 = 81/1681

sin(v) = sqrt(81/1681) ... note: sin(v) > 0 since we're in quadrant II

sin(v) = sqrt(81)/sqrt(1681)

sin(v) = 9/41

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To sum things up so far, we know this much

sin(u) = 11/61 (given)
cos(v) = -40/41 (given)

cos(u) = -60/61 (just found above)
sin(v) = 9/41 (just found above)

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We take these 4 pieces of info and use them in the identity sin(u+v) = sin(u)cos(v) + cos(u)sin(v)

sin(u+v) = sin(u)cos(v) + cos(u)sin(v)

sin(u+v) = (11/61)*(-40/41) + (-60/61)*(9/41)

sin(u+v) = (11*(-40))/(61*41) + (-60*9)/(61*41)

sin(u+v) = (-440)/(2501) + (-540)/(2501)

sin(u+v) = (-440-540)/(2501)

sin(u+v) = -980/2501 <---- Final Answer