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18. (a) Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$ - OCR - GCSE Maths - Question 18 - 2018 - Paper 6

Question 18

$18.-(a)-Sketch-the-graph-of-$y-=-\cos-x-+-1$-for-$0^{\circ}-\leq-x-\leq-720^{\circ}$-OCR-GCSE Maths-Question 18-2018-Paper 6.png$

18. (a) Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$. (b) Explain why the equation $\cos x + 1 = 2.7$ has no solutions.

Worked Solution & Example Answer:18. (a) Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$ - OCR - GCSE Maths - Question 18 - 2018 - Paper 6

Step 1

Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$

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Answer

To sketch the graph of the function $y = \cos x + 1$ , we start by recognizing key features of the cosine function:

Basic Shape: The cosine function oscillates between -1 and 1; therefore, $\cos x + 1$ will oscillate between 0 and 2.
Period: The standard period of $\cos x$ is $360^{\circ}$ , so the behavior of the graph will repeat itself after each $360^{\circ}$ . Hence, when sketching for $0^{\circ} \leq x \leq 720^{\circ}$ , we will sketch two full cycles.
Key Points:
- At $0^{\circ}$ , $y = \cos(0) + 1 = 1 + 1 = 2$ .
- At $90^{\circ}$ , $y = \cos(90) + 1 = 0 + 1 = 1$ .
- At $180^{\circ}$ , $y = \cos(180) + 1 = -1 + 1 = 0$ .
- At $270^{\circ}$ , $y = \cos(270) + 1 = 0 + 1 = 1$ .
- At $360^{\circ}$ , $y = \cos(360) + 1 = 1 + 1 = 2$ .
- The same points repeat at $x = 360^{\circ}$ and $x = 720^{\circ}$ .
Graphing: Start the graph at $(0, 2)$ and follow the points derived above, ensuring that the curve is smooth, reaches its maximum at $x = 0^{\circ}$ , $360^{\circ}$ , and has a minimum at $180^{\circ}$ .
Labeling: Clearly label the x-axis and y-axis with appropriate values, ensuring the range of y covers $0$ to $2$ accordingly.

Step 2

Explain why the equation $\cos x + 1 = 2.7$ has no solutions.

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Answer

The equation $\cos x + 1 = 2.7$ can be rewritten as $\cos x = 2.7 - 1 = 1.7$ .

Range of Cosine: The cosine function, $\cos x$ , has a maximum value of 1 and a minimum value of -1. Thus, it cannot equal any value greater than 1 or less than -1.
Conclusion: Since $1.7$ exceeds the maximum value of $\cos x$ , the equation $\cos x + 1 = 2.7$ has no solutions.

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18. (a) Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$ - OCR - GCSE Maths - Question 18 - 2018 - Paper 6

Worked Solution & Example Answer:18. (a) Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$ - OCR - GCSE Maths - Question 18 - 2018 - Paper 6

Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$

Explain why the equation $\cos x + 1 = 2.7$ has no solutions.

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