Question

If `sin θ = 12/13` then evaluate `((2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ))`.

Answer 1

Given: `sin θ = 12/13`.

Step-wise calculation:

1. From `sin θ = 12/13`, a right triangle gives adjacent = 5, so `cos θ = ±5/13`.

For an acute θ take `cos θ = 5/13`.

2. Numerator: 2 sin θ – 3 cos θ 

= `2 xx (12/13) - 3 xx (5/13)` 

= `(24 - 15)/13`

= `9/13`

3. Denominator: 4 sin θ – 9 cos θ 

= `4 xx (12/13) - 9 xx (5/13)` 

= `(48 - 45)/13`

= `3/13`

4. Ratio: `(9/13)/(3/13)`

= `9/3`

= 3

The value of `((2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)) = 3` for θ in the first quadrant.

If cos θ were `(-5)/13` instead, the same substitution yields `(39/13)/(93/13) = 13/31`, so the sign of cos θ matters; the intended answer is 3.