If sin alpha = -3/5, sin beta = -5/13, P(alpha) is in the third quadrant, and P(beta) is in the fourth quadrant, find the following:

a) sin (alpha - beta)

very similar to your post

http://www.jiskha.com/display.cgi?id=1209665332

To find sin (alpha - beta), we can use the trigonometric identity for the difference of two angles: sin (alpha - beta) = sin alpha * cos beta - cos alpha * sin beta.

Given that sin alpha = -3/5 and sin beta = -5/13, we need to find cos alpha and cos beta.

To find cos alpha, we need to determine the quadrant in which P(alpha) lies. We are given that P(alpha) is in the third quadrant. In the third quadrant, both sine and cosine values are negative.

Using the Pythagorean identity, sin^2 alpha + cos^2 alpha = 1, we can substitute the given value of sin alpha to find cos alpha:

(-3/5)^2 + cos^2 alpha = 1
9/25 + cos^2 alpha = 1
cos^2 alpha = 16/25
cos alpha = -4/5 (taking the negative value since cos alpha is negative in the third quadrant)

Similarly, to find cos beta, we need to determine the quadrant in which P(beta) lies. We are given that P(beta) is in the fourth quadrant. In the fourth quadrant, sine values are negative, but cosine values are positive.

Using the Pythagorean identity again, sin^2 beta + cos^2 beta = 1, we can substitute the given value of sin beta to find cos beta:

(-5/13)^2 + cos^2 beta = 1
25/169 + cos^2 beta = 1
cos^2 beta = 144/169
cos beta = 12/13 (taking the positive value since cos beta is positive in the fourth quadrant)

Now we have sin alpha, sin beta, cos alpha, and cos beta. We can substitute these values into the formula for sin (alpha - beta):

sin (alpha - beta) = sin alpha * cos beta - cos alpha * sin beta
= (-3/5) * (12/13) - (-4/5) * (-5/13)
= -36/65 + 20/65
= -16/65

Therefore, sin (alpha - beta) = -16/65.