SOLUTION: If cosx =4/5 and cos y =12/13,find sin (x+y) when x and y are the measures of two first quadrant angles

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Question 413529: If cosx =4/5 and cos y =12/13,find sin (x+y) when x and y are the measures of two first quadrant angles
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The formula for sin(x+y) is:
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
As you can see, we need sin(x), sin(y, cos(x) and cos(y) for this formula. But you have been given only cos(x) and cos(y). So our first task is to find sin(x) and sin(y).

To find a sin from a cos we will use the identity
sin2(x) = 1 - cos2(x)
Substituting the value we were given for cos(x) we have:
sin2(x) = 1+-+%284%2F5%29%5E2
which simplifies as follows:
sin2(x) = 1+-+16%2F25
sin2(x) = 25%2F25+-+16%2F25
sin2(x) = 9%2F25
Now we find the square root of each side. And since we are told that x is in the first quadrant, we will use only the positive square root:
sin(x) = 3/5

Repeating this for y:
sin2(y) = 1 - cos2(y)
Substituting the value we were given for cos(y) we have:
sin2(y) = 1+-+%2812%2F13%29%5E2
which simplifies as follows:
sin2(y) = 1+-+144%2F169
sin2(y) = 169%2F169+-+144%2F169
sin2(y) = 25%2F169
Now we find the square root of each side. And since we are told that y is in the first quadrant, we will use only the positive square root:
sin(y) = 5/13

Now that we have all four values we can substitute them into the formula for sin(x+y):
sin(x+y) = %283%2F5%29%2812%2F13%29+%2B+%284%2F5%29%285%2F13%29
which simplifies as follows:
sin(x+y) = 36%2F65+%2B+20%2F65
sin(x+y) = 56%2F65