Question

If 5 tan θ = 4, show that `(5 sin θ - 3 cos θ)/(5 sin θ + 2 cos θ) = 1/6`.

Answer 1

Given: 5 tan θ = 4, show that `(5 sin θ - 3 cos θ)/(5 sin θ + 2 cos θ) = 1/6`.

Step-wise calculation:

1. From 5 tan θ = 4, tan θ = `4/5`.

2. Take a right triangle with opposite = 4, adjacent = 5.

So, hypotenuse = `sqrt(4^2 + 5^2) = sqrt(41)`. 

Hence, `sin θ = 4/sqrt(41)` and `cos θ = 5/sqrt(41)`.

3. Compute numerator:

5 sin θ – 3 cos θ 

= `5 xx (4/sqrt(41)) - 3 xx (5/sqrt(41))` 

= `(20 - 15)/sqrt(41)`

= `5/sqrt(41)`

4. Compute denominator:

5 sin θ + 2 cos θ

= `5 xx (4/sqrt(41)) + 2 xx (5/sqrt(41))`

= `(20 + 10)/sqrt(41)`

= `30/sqrt(41)`

5. Ratio:

`(5/sqrt(41))/(30/sqrt(41))`

= `5/30`

= `1/6`

Therefore, `(5 sin θ - 3 cos θ)/(5 sin θ + 2 cos θ) = 1/6`, as required.