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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

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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.png

What is the value of $\tan \alpha$ when the expression $2\sin x - \cos x$ is written in the form $\sqrt{5} \sin(x - \alpha)$?

Worked Solution & Example Answer: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

Step 1

Step 1: Convert the Expression

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Answer

We start with the expression 2sin⁡x−cos⁡x2\sin x - \cos x. Our goal is to express it in the form Rsin⁡(x−α)R \sin(x - \alpha), where R=a2+b2R = \sqrt{a^2 + b^2}, with a=2a = 2 and b=−1b = -1. Thus, we calculate:

R=22+(−1)2=5.R = \sqrt{2^2 + (-1)^2} = \sqrt{5}.

Step 2

Step 2: Find Alpha

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Answer

Next, we determine α\alpha using the relationship:

tan⁡α=ba=−12.\tan \alpha = \frac{b}{a} = \frac{-1}{2}.

Thus, we find:

tan⁡α=−12.\tan \alpha = -\frac{1}{2}.

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