SOLUTION: Find the exact value of the trigonometric function given that sin u = -3/5 and cos v = -7/25 (Both u and v are in Quadrant III.) sin(u+v)

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Question 1179610: Find the exact value of the trigonometric function given that sin u = -3/5 and cos v = -7/25 (Both u and v are in Quadrant III.)
sin(u+v)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Find the exact value of the trigonometric function given that
sin%28+u%29+=+-3%2F5 and
cos+%28v+%29=+-7%2F25
(Both u and v are in Quadrant III.)
first find +cos%28u%29 and sin%28+v%29+
if sin%28+u%29+=+-3%2F5 then opp=3 and hyp=5

use Pythagorean theorem to find adj side
adj=sqrt%285%5E2-3%5E2%29
adj=sqrt%2825-9%29
adj=sqrt%2816%29
adj=4 oradj=+-4
since u+ in Quadrant III, sine and cosine are negative, so use adj=-4
and+cos%28u%29=-4%2F5
if cos+%28v+%29=+-7%2F25 then adj=7 and hyp=25
opp=sqrt%2825%5E2-7%5E2%29
opp=sqrt%28576%29
opp=24 or opp=-24...in Quadrant III sine and cosine are negative
sin%28v%29=-24%2F25

since sin%28+u%29+=+-3%2F5,+cos%28u%29=-4%2F5, sin%28v%29=-24%2F25, cos+%28v+%29=+-7%2F25, then,
sin%28u%2Bv%29+=sin%28u%29+cos%28v%29+%2Bcos%28u%29+sin%28v%29+
sin%28u%2Bv%29+=%28-3%2F5%29+%28-7%2F25%29+%2B++%28-4%2F5%29%28-24%2F25%29
sin%28u%2Bv%29+=21%2F125%2B++96%2F125
sin%28u%2Bv%29+=117%2F125