Cosine graph
Sections:
Questions:
Shape and features of the graph y = a cos bx + c
Question: Deduce the equation of a cosine graph from a diagram
1. The diagram below shows the graph of $y = a \cos (bx) + c $.
Determine the values of $a$, $b$ and $c$.
(from A Maths 360 Workbook Ex 11.3)
Answer: $ a = 3, b = 2, c = 0 $
2. The diagram below shows the graph of $y = a \cos (bx) + c $.
Determine the values of $a$, $b$ and $c$.
(from A Maths 360 Workbook Ex 11.3)
Answer: $ a = -2, b = 4, c = 2 $
Question: Deduce the equation of a cosine graph from it's features
3. The curve $y = a \cos \left(x \over b\right) + c$, where $a$ and $b$ are positive integers, has an amplitude of $4$ units and period of $720^\circ$. The maximum value of $y$ is $3$. State the values of $a$, $b$ and $c$.
Answer: $ a = 4, b = 2, c = -1 $
Question: Sketch a cosine graph
4. Sketch the graph of $y = -1 - \cos \left({x \over 2}\right)$, for $ 0 \le x \le 6 \pi $.
Question: Symmetry of a cosine graph
5. The diagram above shows part of the graph of $y = 6 - 3 \cos 2x$, passing through the points $(k, 4)$, $(l, 8)$ and $(m, 4)$, where $k$, $l$ and $m$ are constants.
Using symmetry of the graph, or otherwise, find an equation connecting
(from think! A Maths Worksheet 9B)
(i) $\pi$, $k$ and $m$,
Answer: $ m = k + \pi $
(ii) $\pi$, $k$ and $l$,
Answer: $ k + l = {\pi \over 2} $
Question: Use a cosine graph to deduce the number of solutions to an equation
6(i) On the same axes, sketch the graphs of $y = 2 \cos 2x$ and $y = \sin x + 1$ for $0 \le x \le 2\pi$.
(from A Maths 360 2nd edition Ex 10.2)
6(ii) Hence, find the number of distinct values of $x$, in the interval $0 \le x \le 2\pi$, for which $\cos 2x - {1 \over 2} = {1 \over 2} \sin x $.
Answer: $ \text{4 distinct values of } x $
O Level past year questions on the cosine graph
3. The curve $y = a \cos \left(x \over b\right) + c$, where $a$ and $b$ are positive integers, has an amplitude of $4$ units and period of $720^\circ$. The maximum value of $y$ is $3$. State the values of $a$, $b$ and $c$.
Answer: $ a = 4, b = 2, c = -1 $
4. Sketch the graph of $y = -1 - \cos \left({x \over 2}\right)$, for $ 0 \le x \le 6 \pi $.
5. The diagram above shows part of the graph of $y = 6 - 3 \cos 2x$, passing through the points $(k, 4)$, $(l, 8)$ and $(m, 4)$, where $k$, $l$ and $m$ are constants.
Using symmetry of the graph, or otherwise, find an equation connecting
(from think! A Maths Worksheet 9B)
(i) $\pi$, $k$ and $m$,
Answer: $ m = k + \pi $
(ii) $\pi$, $k$ and $l$,
Answer: $ k + l = {\pi \over 2} $
6(i) On the same axes, sketch the graphs of $y = 2 \cos 2x$ and $y = \sin x + 1$ for $0 \le x \le 2\pi$.
(from A Maths 360 2nd edition Ex 10.2)
6(ii) Hence, find the number of distinct values of $x$, in the interval $0 \le x \le 2\pi$, for which $\cos 2x - {1 \over 2} = {1 \over 2} \sin x $.
Answer: $ \text{4 distinct values of } x $
| Year & paper | Comments |
|---|---|
| 2025 P1 Question 12 | Geometry question (very different from prior years) |
| 2024 P1 Question 10 | Sketch graph and deduce solutions to equation |
| 2023 P2 Question 7a | Deduce equation from information provided (difficult) |
| 2022 P1 Question 10 | Sketch graph |
| 2017 P2 Question 10 | Sketch graph and deduce number of solutions to inequality |
| 2016 P1 Question 3b | Deduce equation from graph |
| 2014 P2 Question 9 | Sketch graph and deduce solutions to equation |
| 2012 P1 Question 8 | Deduce equation from information and sketch graph |
| 2010 P2 Question 11i | Deduce equation from information and sketch graph |
| 2008 P2 Question 7 | Sketch graph |
| 2007 P1 Question 9 | Sketch graph and deduce solutions to equation |
| 2004 P1 Question 6 | Find amplitude, period and the coordinates of maximum point and minimum point (Link - Subscription required) |