What is the value of tan \( \alpha \) when the expression \( 2 \sin x - \cos x \) is written in the form \( \sqrt{5} \sin(x - \alpha) \)? - HSC - SSCE Mathematics Extension 1 - Question 4 - 2017 - Paper 1

We can express ( 2 \sin x - \cos x ) as follows:

1. Identify coefficients: ( R = \sqrt{2^2 + (-1)^2} = \sqrt{5} )
2. Write ( \sin(x - \alpha) ) using the angle addition formula: [ R \sin(x - \alpha) = R (\sin x \cos \alpha - \cos x \sin \alpha) ] Matching coefficients:
   [ \sqrt{5} \cos \alpha = 2 \quad\text{and}\quad \sqrt{5} \sin \alpha = -1 ]

Using the definitions of sine and cosine, calculate ( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} ):

Substituting in the values we have: [ \tan \alpha = \frac{-1 / \sqrt{5}}{2 / \sqrt{5}} = \frac{-1}{2} ]