What is the value of tan α when the expression 2sin x − cos x is written in the form √5 sin(x − α)? - HSC - SSCE Mathematics Extension 1 - Question 4 - 2017 - Paper 1

To write the expression in the form of (\sqrt{5}\text{sin}(x - \alpha)), we can use the identity:

$R \text{sin}(x - \alpha) = R \text{sin} x \text{cos} \alpha - R \text{cos} x \text{sin} \alpha$

Comparing coefficients, we get:

• The coefficient of sin x is 2, which means we have (R \cos \alpha = 2)
• The coefficient of cos x is -1, which gives us (-R \sin \alpha = -1) or (R \sin \alpha = 1)

By Pythagorean identities, we can find (R):

$R = \sqrt{(R \cos \alpha)^2 + (R \sin \alpha)^2}$ Substituting the values:

$R = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$

We have:

$\tan \alpha = \frac{R \sin \alpha}{R \cos \alpha} = \frac{1}{2}$

Thus, the value of (\tan \alpha) is (\frac{1}{2}). Therefore, the answer is C.