If sin theta is 8÷17;cos beta is 9÷41 where theta is obtuse and beta is acute. Find sin(theta + beta)
We have angle a in QII, and angle b in QI. So,
sin a = 8/17
cos a = -15/17
sin b = 40/41
cos b = 9/41
sin(a+b) = sin a * cos b + cos a * sin b
= (8/17)(9/41)+(-15/17)(40/41) = -528/697
To find the value of sin(theta + beta), you can use the trigonometric identity:
sin(theta + beta) = sin(theta) * cos(beta) + cos(theta) * sin(beta)
Given that sin(theta) = 8/17 and cos(beta) = 9/41, let's find the value of cos(theta) and sin(beta) first.
To find cos(theta), we can use the Pythagorean Identity:
sin^2(theta) + cos^2(theta) = 1
Substituting sin(theta) = 8/17, we can solve for cos(theta):
(8/17)^2 + cos^2(theta) = 1
64/289 + cos^2(theta) = 1
cos^2(theta) = 225/289
cos(theta) = sqrt(225/289)
Since theta is obtuse, cos(theta) will be negative:
cos(theta) = - sqrt(225/289)
cos(theta) = -15/17
Next, let's find sin(beta). We can use the Pythagorean Identity again:
sin^2(beta) + cos^2(beta) = 1
Substituting cos(beta) = 9/41, we can solve for sin(beta):
sin^2(beta) + (9/41)^2 = 1
sin^2(beta) = 1 - (9/41)^2
sin(beta) = sqrt(1 - (9/41)^2)
Now plug in the values we found into the expression for sin(theta + beta):
sin(theta + beta) = sin(theta) * cos(beta) + cos(theta) * sin(beta)
= (8/17) * (9/41) + (-15/17) * sqrt(1 - (9/41)^2)
Evaluating this expression will give you the value of sin(theta + beta).