Question

If `"cosec"  θ = 13/12`, find the value of `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)`

Answer 1

Given: cosec θ = `13/12` `("so" sin θ = 12/13)`.

To Prove: `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)`.

Proof [Step-wise]:

1. From cosec θ = `13/12` we get `sin θ = 12/13`.

2. Use sin² θ + cos² θ = 1 to find cos θ:

`sin^2 θ = (12/13)^2 = 144/169`

So `cos^2 θ = 1 - 144/169 = 25/169`. 

Hence `cos θ = ±5/13`.   ...(Sign depends on quadrant of θ.)

3. Compute the numerator and denominator with `sin θ = 12/13` and `cos θ = 5/13` (cos positive):

Numerator = 2 sin θ – 3 cos θ

= `2 xx (12/13) - 3 xx (5/13)` 

= `(24 - 15)/13`

= `9/13`

Denominator = 4 sin θ – 9 cos θ

= `4 xx (12/13) - 9 xx (5/13)` 

= `(48 - 45)/13`

= `3/13`

Ratio = `(9/13)/(3/13)` 

= `9/3`

= 3

4. Compute with `cos θ = -5/13` (cos negative):

Numerator = `2 xx (12/13) - 3 xx (-5/13)` 

= `(24 + 15)/13`

= `39/13`

Denominator = `4 xx (12/13) - 9 xx (-5/13)` 

= `(48 + 45)/13`

= `93/13`

Ratio = `(39/13)/(93/13)`   ...(After dividing numerator and denominator by 3)

= `39/93`

= `13/31` 

5. Therefore the expression has two possible values depending on the sign of cos θ:

If θ is in quadrant I (sin > 0, cos > 0) → value = 3.

If θ is in quadrant II (sin > 0, cos < 0) → value = `13/31`.

The value of `(2 sin θ - 3 cos θ)/(4 sin θ - 9 cos θ)` is 3 if `cos θ = +5/13` and `13/31` if `cos θ = −5/13`. The value is not uniquely determined by cosec θ = `13/12` alone the quadrant must be specified.