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Express $7 \, \cos x - 24 \, \sin x$ in the form $R \cos(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 4
Question 3
Express $7 \, \cos x - 24 \, \sin x$ in the form $R \cos(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$. Give the value of $\alpha$ to 3 decimal places.... show full transcript
Worked Solution & Example Answer:Express $7 \, \cos x - 24 \, \sin x$ in the form $R \cos(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$ - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 4
Step 1
Express $7\cos x - 24 \sin x$ in the form $R \cos(x + \alpha)$
Answer
We start by letting [ R = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25. ]
Now, we find ( \tan(\alpha) ):
[ \tan(\alpha) = \frac{-24}{7}. ]
Therefore, [ \alpha = \tan^{-1}\left(-\frac{24}{7}\right) \approx -1.287. ]
Since we need a positive angle, we will add .
Therefore, [ \alpha \approx 1.287 + \pi \approx 1.287 + 3.142 = 4.429. ]
Thus, we can write [ 7\cos x - 24\sin x = 25\cos\left(x + 1.287\right). ]
Step 2
Step 3
Solve, for $0 \leq x < 2\pi$, the equation $7 \, \cos x - 24 \, \sin x = 10$
Answer
We can rewrite the equation as [ 25\cos(x + 1.287) = 10. ]
Dividing both sides by 25 gives:
[ \cos(x + 1.287) = \frac{10}{25} = 0.4. ]
Now, taking the inverse cosine we find:
[ x + 1.287 = \cos^{-1}(0.4). ]
Calculating gives:
[ x + 1.287 \approx 1.159 \text{ and } x + 1.287 \approx -1.159 + 2\pi ].
The corresponding values will then be calculated as follows:
- [ x = 1.159 - 1.287 \approx -0.128 \text{ (not valid since } x \geq 0) ]
- [ x = 2\pi - 1.159 + 1.287 \approx 6.282. ]
Thus, x values are approximately:
[ 3.84 \text{ and } 6.16 \text{ (to 2 decimal places).} ]