SOLUTION: cos a = 4/9, a in Q1, and tan b = -5/4, b in Q2. Find sin (a+b), cos (a+b), tan (a+b)

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Question 927606: cos a = 4/9, a in Q1, and tan b = -5/4, b in Q2.
Find sin (a+b), cos (a+b), tan (a+b)
Found 2 solutions by Alan3354, KMST:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
cos a = 4/9, a in Q1, and tan b = -5/4, b in Q2.
Find sin (a+b), cos (a+b), tan (a+b)
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sin(a+b) = sin(a)cos(b) + sin(b)cos(a)
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cos(a+b) = cos*cos - sin*sin
---------
tan = sin/cos

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If we need exact values, we need to use the trigonometric identities
cos%28A%2BB%29=cos%28A%29%2Acos%28B%29-sin%28A%29%2Asin%28B%29 and sin%28A%2BB%29=cos%28A%29%2Asin%28B%29%2Bsin%28A%29%2Acos%28B%29 ,
so as a first step, we are going to find exact values for sin%28A%29 and cos%28b%29 .

cos%28A%29=4%2F9 in Q1.
In quadrant 1, sine and cosine are positive , and we know that %28cos%28A%29%29%5E2%2B%28sin%28A%29%29%5E2=1 , so


tan%28B%29=-5%2F4 in Q2 means
From the red right rectangle, we calculate
sin%28B%29=5%2Fsqrt%2841%29=5sqrt%2841%29%2F41 and cos%28B%29=-4%2Fsqrt%2841%29=-4sqrt%2841%29%2F41 .

Now,


tan%28A%2BB%29=sin%28A%2BB%29%2Fcos%28A%2BB%29===

=-4%2A41%2A%2880-25sqrt%2865%29-16sqrt%2865%29%2B5%2A65%29%2F%2841%2A%28256-25%2A65%29%29=-4%2A%2880-25sqrt%2865%29-16sqrt%2865%29%2B5%2A65%29%2F%28256-1625%29=-4%2A%2880-41sqrt%2865%29%2B325%29%2F%28256-1625%29

=-4%2A%28405-41sqrt%2865%29%29%2F%28-1369%29=highlight%281620-164sqrt%2865%29%29%2F1369%29