Given:a and b are first-quadrant angles, sin a = 5/13 , and cos b = 3/5 , evaluate sin(a - b).
To evaluate sin(a - b), we can use the trigonometric identity for the difference of angles:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Given that sin a = 5/13 and cos b = 3/5, we can substitute these values into the formula:
sin(a - b) = (5/13)(3/5) - cos(a)sin(b)
To find cos(a), we need to use the Pythagorean identity:
cos^2(a) = 1 - sin^2(a)
Since a is a first-quadrant angle, cos(a) is positive. We can solve for cos(a) as follows:
cos^2(a) = 1 - sin^2(a)
cos^2(a) = 1 - (5/13)^2
cos^2(a) = 1 - 25/169
cos^2(a) = (169 - 25) / 169
cos^2(a) = 144 / 169
Taking the square root of both sides, we get:
cos(a) = √(144 / 169)
cos(a) = 12 / 13
Now we substitute the values of cos(a) and sin(b) into the formula for sin(a - b):
sin(a - b) = (5/13)(3/5) - (12/13)(sin(b))
sin(a - b) = 15/65 - (12/13)(sin(b))
To evaluate sin(b), we can use the sine formula:
sin^2(b) = 1 - cos^2(b)
Since b is a first-quadrant angle, sin(b) is positive. We can solve for sin(b) as follows:
sin^2(b) = 1 - cos^2(b)
sin^2(b) = 1 - (3/5)^2
sin^2(b) = 1 - 9/25
sin^2(b) = (25 - 9) / 25
sin^2(b) = 16 / 25
Taking the square root of both sides, we get:
sin(b) = √(16 / 25)
sin(b) = 4 / 5
Now we substitute the values of cos(a), sin(b), and sin(a - b) into the formula:
sin(a - b) = 15/65 - (12/13)(4/5)
sin(a - b) = 15/65 - (48/65)
Finally, simplify the expression:
sin(a - b) = -33/65
Therefore, sin(a - b) is equal to -33/65.