The value of a are ±5, and b is √61.
To determine the values of a and b in the expression sinθ = a/b, given that tanθ = 5/6, we can use the fundamental trigonometric identity tanθ = sinθ/cosθ.
Since tanθ = 5/6, we can write:
(5/6) = (sinθ)/(cosθ)
Now, using the Pythagorean identity sin^2θ + cos^2θ = 1, we know that cos^2θ = 1 - sin^2θ.
Substitute this into the expression for tanθ:
(5/6) = (sinθ)/√(1 - sin^2θ)
Cross-multiply to get rid of the fraction:
5√(1 - sin^2θ) = 6sinθ
Square both sides to eliminate the square root:
25(1 - sin^2θ) = 36sin^2θ
Simplify:
25 - 25sin^2θ = 36sin^2θ
Combine like terms:
61sin^2θ = 25
Solve for sinθ:
sinθ = ±5/√61
Now, compare this with the expression sinθ = a/b:
a/b = ±5/√61
Therefore, the values of a and b are a = ±5 and b = √61.