Begin with:
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
Since sin(x) = 7/25, we can square both sides of the equation to find sin^2(x):
sin^2(x) = (7/25)^2 = 49/625
Next, we can subtract sin^2(x) from 1 to find cos^2(x):
cos^2(x) = 1 - sin^2(x)
cos^2(x) = 1 - 49/625
cos^2(x) = (625 - 49)/625
cos^2(x) = 576/625
Taking the square root of both sides gives us the value of cos(x):
cos(x) = ±√(576/625)
Since x is between 0 and π/2, cos(x) is positive.
cos(x) = √(576/625)
cos(x) = 24/25
Now, we can use the given value of sec(y) = 13/12 to find the value of cos(y). Recall that sec(y) = 1/cos(y), so we can write:
1/cos(y) = 13/12
Cross-multiplying:
12 = 13cos(y)
cos(y) = 12/13
Since cos(y) is positive, y must be between 0 and π/2.
Now, we have the values of sin(x) = 7/25, cos(x) = 24/25, and cos(y) = 12/13. To find sin(x + y), we can use the identity:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Substituting the given values:
sin(x + y) = (7/25)(12/13) + (24/25)(√(1 - (12/13)^2))
Simplifying further:
sin(x + y) = (84/325) + (24/25)(√(1 - 144/169))
sin(x + y) = (84/325) + (24/25)(√(25/169))
Looking at the term (√(25/169)), we can simplify it further:
√(25/169) = 5/13
Substituting back into the equation:
sin(x + y) = (84/325) + (24/25)(5/13)
sin(x + y) = (84/325) + (24/65)
sin(x + y) = (84/325) + (48/325)
sin(x + y) = 132/325
Therefore, the exact value of sin(x + y) is 132/325, where p = 132 and q = 325.