Final answer:
To find sin(a-b), substitute the values of sin a and sin b into the trigonometric identity sin(a-b) = sin a cos b - cos a sin b.
Step-by-step explanation:
Given that cos a = 54 with a in quadrant 1, we can find sin a using the Pythagorean identity: sin^2 a + cos^2 a = 1. Rearranging the equation, sin^2 a = 1 - cos^2 a. Substituting the value of cos a, we have sin^2 a = 1 - 54^2. Taking the square root of both sides, sin a = √(1 - 54^2).
Similarly, given tan b = 512 with b in quadrant 3, we can find sin b using the Pythagorean identity: sin^2 b = 1 - tan^2 b. Substituting the value of tan b, we have sin^2 b = 1 - 512^2. Taking the square root of both sides, sin b = -√(1 - 512^2) (since b is in quadrant 3).
To find sin(a-b), we can use the trigonometric identity: sin(a-b) = sin a cos b - cos a sin b. Substituting the values we found earlier, we have sin(a-b) = (√(1 - 54^2))(√(1 - 512^2)) - (54)(-√(1 - 512^2)). Simplifying this expression will give us the value of sin(a-b).