The graphs of $y = \sin x$ and $y = \cos x$ are among the most important in mathematics. They model everything from sound waves to planetary orbits. In this lesson, you will learn the key features of these graphs — amplitude, period, and intercepts — and discover how to transform them into more general sinusoidal functions.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
You know that $\sin 0 = 0$, $\sin \frac{\pi}{2} = 1$, $\sin \pi = 0$, $\sin \frac{3\pi}{2} = -1$, and $\sin 2\pi = 0$. If you plot these points and join them with a smooth curve, what shape do you expect to get? And how do you think the graph of $y = \cos x$ will differ?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: √(a + b) = √a + √b.
Right: The square root of a sum is not the sum of square roots; √(a+b) cannot be simplified this way.
📚 Core Content
The graph of $y = \sin x$ is a smooth wave that:
Amplitude: $1$ Period: $2\pi$ Range: $[-1, 1]$
The graph of $y = \cos x$ is the same wave shape, but shifted horizontally:
Amplitude: $1$ Period: $2\pi$ Range: $[-1, 1]$
For functions of the form $y = a\sin(bx) + d$ and $y = a\cos(bx) + d$:
| Parameter | Effect |
|---|---|
| $a$ | Amplitude = $|a|$. If $a < 0$, the graph is reflected in the $x$-axis. |
| $b$ | Period = $\frac{2\pi}{|b|}$ (radians) or $\frac{360^\circ}{|b|}$ (degrees). If $|b| > 1$, the graph is compressed horizontally. If $0 < |b| < 1$, it is stretched. |
| $d$ | Vertical shift. The midline of the wave moves from $y = 0$ to $y = d$. |
The maximum value of $a\sin(bx) + d$ is $|a| + d$, and the minimum value is $-|a| + d$. Therefore:
$$\text{Range: } [d - |a|, \, d + |a|]$$
🧮 Worked Examples
🧪 Activities
1 $y = \sin x$
Type key features:
Sketch in your workbook.
2 $y = 2\cos x$
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3 $y = \sin(2x)$
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4 $y = \cos x + 2$
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1 $y = 4\sin x$
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Answer in your workbook.
2 $y = \cos(3x)$
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3 $y = 2\sin x - 1$
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Answer in your workbook.
Earlier you were asked to predict the shape of the sine and cosine graphs.
The points $(0, 0)$, $\left(\frac{\pi}{2}, 1\right)$, $(\pi, 0)$, $\left(\frac{3\pi}{2}, -1\right)$, $(2\pi, 0)$ form a smooth wave — the sine curve. The cosine curve uses $(0, 1)$, $\left(\frac{\pi}{2}, 0\right)$, $(\pi, -1)$, $\left(\frac{3\pi}{2}, 0\right)$, $(2\pi, 1)$, giving the same wave shape shifted left by $\frac{\pi}{2}$.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. Sketch $y = 3\cos(2x)$ for $0 \leq x \leq 2\pi$. Label the amplitude, period, and the coordinates of all maximum and minimum points. 4 MARKS
Describe your sketch below:
Sketch in your workbook.
9. Find the exact period and range of $y = 2\sin\left(\frac{x}{2}\right) + 1$. 3 MARKS
Type your answer below:
Answer in your workbook.
10. Explain why the graph of $y = \cos x$ can be obtained from the graph of $y = \sin x$ by a horizontal translation. State the exact size and direction of this translation. 3 MARKS
Type your answer below:
Answer in your workbook.
1. Standard sine wave: amplitude 1, period $2\pi$, passes through origin.
2. Amplitude 2, period $2\pi$, starts at $(0, 2)$.
3. Amplitude 1, period $\pi$, passes through $(0, 0)$, max at $\frac{\pi}{4}$.
4. Amplitude 1, period $2\pi$, shifted up 2, oscillates between $y = 1$ and $y = 3$.
1. Amplitude = 4, Period = $2\pi$, Range = $[-4, 4]$
2. Amplitude = 1, Period = $\frac{2\pi}{3}$, Range = $[-1, 1]$
3. Amplitude = 2, Period = $2\pi$, Range = $[-3, 1]$
1. A — $y = \sin x$ has period $2\pi$.
2. A — Amplitude of $3\cos x$ is $3$.
3. A — Period of $\cos(2x)$ is $\pi$.
4. A — Range of $2\sin x + 1$ is $[-1, 3]$.
5. A — $\cos x = \sin\left(x + \frac{\pi}{2}\right)$.
Q8 (4 marks): Amplitude = 3 [0.5], Period = $\pi$ [0.5]. Max points at $(0, 3)$, $(\pi, 3)$, $(2\pi, 3)$ [1.5]. Min points at $\left(\frac{\pi}{2}, -3\right)$, $\left(\frac{3\pi}{2}, -3\right)$ [1.5].
Q9 (3 marks): Period = $\frac{2\pi}{\frac{1}{2}} = 4\pi$ [1]. Amplitude = 2 [0.5]. Range = $[1 - 2, 1 + 2] = [-1, 3]$ [1.5].
Q10 (3 marks): $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ [1]. This means replacing $x$ with $x + \frac{\pi}{2}$ in $y = \sin x$ [1]. This corresponds to a horizontal translation of $\frac{\pi}{2}$ units to the left [1].
Challenge the boss using your knowledge of graphing sine and cosine functions. Pool: lessons 1–10.
Tick when you've finished all activities and checked your answers.