Question

Two wires support a utility pole and form angles α and β with the ground. Find the value of sin(α - β) if tan α=4/3 on the interval (0°,90°) and cotβ=5/12 on the interval (0°,90°)?

Please show work/explain :)

Answer 1

Given: `tan(alpha) = 4/3` and `cot(beta) = 5/12`

If `cot(beta) = 5/12`, then `tan(beta) = 12/5`

Using the identity `tan(alpha-beta) = (tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta)`

`tan(alpha-beta) = (4/3-12/5)/(1+(4/3)(12/5)`

`tan(alpha-beta) = -16/63`

Using the identity `sec(alpha-beta)= +-sqrt(1+tan^2(alpha-beta))`

`sec(alpha-beta) = +-sqrt(1+(-16/63)^2)`

Because we are told that `alpha` and `beta` are in the first quadrant and we observe that `tan(alpha-beta)` is negative, we conclude that `alpha-beta` is in the fourth quadrant and, therefore, the secant is positive:

`sec(alpha-beta) = 65/63`

Using the identity `sec(theta) = 1/cos(theta)`

`cos(alpha-beta) = 63/65`

Using the identity `tan(alpha-beta) = sin(alpha-beta)/cos(alpha-beta)`

`sin(alpha-beta) = -16/63 63/65`

`sin(alpha-beta) = -16/65`