SOLUTION: 1. Given sinB=5/13 in QII and (6,-8) is on the terminal side of a ,find the exact value of sin (a+B).

Question 1174989: 1. Given sinB=5/13 in QII and (6,-8) is on the terminal side of a ,find the exact value of sin (a+B).
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

If sin(B) = 5/13, then cos(B) = -12/13. Use the pythagorean theorem to see this. Draw out a right triangle. Note how cosine is negative in quadrant 2.

The distance from (0,0) to (6,-8) is 10 units. Again you'll need the pythagorean theorem to see why this is the case, or you could use the distance formula (which is a rephrasing of the pythagorean theorem).

We have a right triangle with hypotenuse 10, and the two legs are 6 and 8 units each. Let's say we had a third point at (6,0). This is where the 90 degree angle is located. The distance from (0,0) to (6,0) is 6 units to form the horizontal leg, while the vertical leg is 8 units because it spans from (6,0) to (6,-8)

So,
sin(a) = opposite/hypotenuse
sin(a) = -8/10
sin(a) = -4/5
and
cos(a) = adjacent/hypotenuse
cos(a) = 6/10
cos(a) = 3/5
In quadrant 4, sine is negative and cosine is positive

--------------------------------------------------

We have these four facts
sin(a) = -4/5
cos(a) = 3/5
sin(B) = 5/13
cos(B) = -12/13

Now we use a trig identity to find what we're after
sin(a+B) = sin(a)*cos(B) + cos(a)*sin(B)
sin(a+B) = (-4/5)*(-12/13) + (3/5)*(5/13)
sin(a+B) = 48/65 + 15/65
sin(a+B) = (48+15)/65
sin(a+B) = 63/65

Answer: 63/65

RELATED QUESTIONS
Given sin B=5/13 in QII,and(6,-8) is on the terminal side of a, find the exact value of... (answered by math_tutor2020,greenestamps)

Given sin β = 5/13 in QII, and (6,-8) is on the terminal side of α, find the exact... (answered by greenestamps,ikleyn)

1. Given sinB=5/13 in QII and (6,-8) is on the terminal side 2.Given SinB = 5/13 ,find... (answered by ikleyn)

Given sin β 5/13 in QII, find the exact value of the following: a. sin(2β) b.... (answered by mananth)

if cosA=(-7/25) and sinB=(4/5), with A in QIII and B in QII, find the exact value of the... (answered by lwsshak3)

Let sin(A) = 3/5 with A in QII. What is the exact value of sin(2A) using an... (answered by lwsshak3)

Given a point (5,-5) on the terminal side of angle0, find the value of... (answered by HyperBrain)

Given: angles A and B are in Quadrant I, where tanA = 1/7 sinB= 5/SQRT74 Find the... (answered by lwsshak3)

let (-3,5) be a point on the terminal side of 0. find the exact value of Sin,Csc, and Cot (answered by Alan3354)