Answer: We can use the trigonometric identity cos(A - B) = cosA cosB + sinA sinB and the given value of cos(A - B) to find the value of cosA cosB - sinA sinB. Then, we can use the given value of secA to find the value of cosA, and use the fact that secA = 1/cosA to find the value of sinA. Finally, we can solve for sinB using the equation we obtained earlier.
cos(A - B) = cosA cosB + sinA sinB
84/85 = (17/8) cosB + sinA sinB
We can rewrite the expression (17/8) cosB + sinA sinB as (17/8) (cosB + (1/17) sinA sinB), and then square both sides of the equation to obtain:
(84/85)^2 = (17/8)^2 (cosB + (1/17) sinA sinB)^2
Simplifying, we get:
(cosB + (1/17) sinA sinB)^2 = (84/85)^2 / (17/8)^2
cosB + (1/17) sinA sinB = ± (84/85) / (17/8)
cosB + (1/17) sinA sinB = ± 32/85
Now, we can substitute the values of secA and cos(A - B) to obtain:
cosA = 1/secA = 8/17
cosA cosB - sinA sinB = cos(A - B) = 84/85
Substituting these values into the expression we obtained earlier, we get:
(8/17) cosB - sinA sinB = 84/85
Multiplying both sides by 17/8, we get:
cosB - (17/8) sinA sinB = (17/8) (84/85)
Adding this equation to the previous equation, we get:
2 cosB = (17/8) (84/85) + (32/85)
Solving for cosB, we get:
cosB = [(17/8) (84/85) + (32/85)] / 2 = 83/85
Using the identity sin^2(A) + cos^2(A) = 1 and the fact that secA = 1/cosA, we can find the value of sinA:
sin^2(A) + cos^2(A) = 1
sin^2(A) = 1 - cos^2(A)
sinA = ±sqrt(1 - cos^2(A))
sinA = ±sqrt(1 - (8/17)^2) = ±15/17
Since secA = 17/8, we know that cosA is positive, and therefore sinA must also be positive. Therefore, sinA = 15/17.
Now, we can use the equation we obtained earlier to find the value of sinB:
cosB - (17/8) sinA sinB = (17/8) (84/85)
Substituting the values we obtained for cosB, sinA, and the given value of secA, we get:
83/85 - (17/8) (15/17) sinB = (17/8) (84/85)
Simplifying, we get:
sinB = [(83/85) - (17/8) (15/17) (85/84)] / [(17/8) (85/84)]
sinB = -24/85
Therefore, the value of sinB is -24/85.
Explanation: