Sin(theta + phi) is evaluated as (-12√5/65) - 5/13 * √5/5 with sin(theta) = 12/13 in Quadrant I and cos(phi) = -√5/5 in Quadrant II.
To evaluate the expression sin(theta + phi) under the given conditions, we can use trigonometric identities to express sin(theta + phi) in terms of sin(theta), cos(phi), and their respective angles.
The sum of angles identity for sine is sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Applying this to sin(theta + phi), we substitute sin(theta) = 12/13 and cos(phi) = -√5/5:
sin(theta + phi) = sin(theta)cos(phi) + cos(theta)sin(phi)
= (12/13 * -√5/5) + (√(1 - (12/13)^2) * ( -√5/5))
Now, since theta is in Quadrant I and phi is in Quadrant II, we know that sin(theta) and cos(phi) are positive. Therefore, we use the positive square root for sin(phi) in the expression.
Calculating further, we find:
= (-12√5/65) - (√(1 - (12/13)^2) * √5/5)
Simplifying and using the fact that sin(phi) is positive in Quadrant II, we get:
= (-12√5/65) - (1 - 12^2/13^2)^(1/2) * √5/5
= (-12√5/65) - (1 - 144/169)^(1/2) * √5/5
= (-12√5/65) - (25/169)^(1/2) * √5/5
= (-12√5/65) - 5/13 * √5/5
Therefore, sin(theta + phi) is equal to (-12√5/65) - 5/13 * √5/5 under the given conditions.
The question probable may be:
Evaluate the expression under the given conditions. sin(theta + phi); sin(theta) = 12 / 13, theta in Quadrant I, cos (phi) = - square root 5 / 5, phi in Quadrant II.