Graphs of Sine and Cosine
The graphs of $y = \sin x$ and $y = \cos x$ model everything from sound waves to planetary orbits. In this lesson you will learn the key features — amplitude, period, and intercepts — and discover how to transform these graphs into more general sinusoidal functions.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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? And how do you think the graph of $y = \cos x$ will differ?
Every sine or cosine graph you will meet in this course can be written in one of these two forms. Learn to read the parameters and you can sketch any transformed wave in under a minute.
The general forms are $y = a\sin(bx) + d$ and $y = a\cos(bx) + d$. The parameter $a$ controls amplitude (and reflects if negative), $b$ controls horizontal stretching via the period formula, and $d$ shifts everything vertically. The range is always centred on the midline $y = d$ and extends $|a|$ units above and below.
Key facts
- The shape and key features of $y = \sin x$ and $y = \cos x$
- How to find amplitude, period, and vertical shift from an equation
- The relationship between degrees and radians in graphing
Concepts
- Why sine and cosine are periodic with period $2\pi$
- How the parameter $b$ affects horizontal stretching/compressing
- Why the cosine graph is a horizontal translation of the sine graph
Skills
- Sketch the graphs of $y = \sin x$ and $y = \cos x$
- Sketch transformed sine and cosine graphs
- Find amplitude, period, and range from an equation
- Read key features from a graph
The graph of $y = \sin x$ is a smooth wave that passes through the origin, reaches a maximum of $1$ at $x = \frac{\pi}{2}$, returns to $0$ at $x = \pi$, hits a minimum of $-1$ at $x = \frac{3\pi}{2}$, and completes one full cycle at $x = 2\pi$.
The graph of $y = \cos x$ has the identical wave shape, but starts at a maximum of $1$ when $x = 0$. It crosses zero at $x = \frac{\pi}{2}$, reaches $-1$ at $x = \pi$, and returns to $1$ at $x = 2\pi$.
Sine (blue) and cosine (orange) waves — both have period $2\pi$, amplitude 1, and are phase-shifted by $\frac{\pi}{2}$
$y = \sin x$: starts at 0, max 1 at $\frac{\pi}{2}$, returns to 0 at $\pi$, min $-1$ at $\frac{3\pi}{2}$, back to 0 at $2\pi$; $y = \cos x$: starts at 1, drops to 0 at $\frac{\pi}{2}$, min $-1$ at...
Pause — copy the five key points of $y = \sin x$ (0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, $2\pi$) and the fact that $y = \cos x$ starts at 1 (same shape shifted $\frac{\pi}{2}$ left) into your book.
Did you get this? True or false: the graph of $y = \cos x$ starts at 1 when $x = 0$, whereas $y = \sin x$ starts at 0.
We just saw that $y = \sin x$ and $y = \cos x$ each complete one cycle in $2\pi$ and oscillate between $-1$ and $1$. That raises a question: how does the graph change if we scale the height, change the speed of oscillation, or shift the midline? This card answers it → the general form $y = a\sin(bx) + d$: amplitude $= |a|$, period $= \frac{2\pi}{|b|}$, midline $y = d$.
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 $y = d$. |
Finding the Range
The maximum value of $a\sin(bx) + d$ is $|a| + d$, and the minimum value is $-|a| + d$. Therefore:
$$\text{Range: } [d - |a|, \, d + |a|]$$
General form: $y = a\sin(bx) + d$ or $y = a\cos(bx) + d$; Amplitude = $|a|$; Period = $\frac{2\pi}{|b|}$; midline = $y = d$
Pause — copy the general form $y = a\sin(bx) + d$ with amplitude $= |a|$, period $= \frac{2\pi}{|b|}$, and midline $y = d$ into your book.
Quick check: What is the period of $y = \cos(3x)$?
Worked examples · 3 in a row, reveal as you go
Sketch one cycle of $y = 2\sin x$ for $0 \leq x \leq 2\pi$ and state its amplitude and range.
For $y = 3\cos(2x)$, find the amplitude, period (in radians), and range.
We just saw worked examples for amplitude scaling and period compression. That raises a question: what happens when we add a constant $d$ to the function — how does a vertical shift affect the key points we need to plot? This card answers it → shift every $y$-value up or down by $d$; the shape is identical but the midline moves from $y = 0$ to $y = d$.
Sketch $y = \sin x + 1$ for $0 \leq x \leq 2\pi$ and state the range.
$y = 2\sin x$: amplitude 2, period $2\pi$, key points $(0,0)$, $(\frac{\pi}{2}, 2)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -2)$, $(2\pi, 0)$; $y = 3\cos(2x)$: amplitude 3, period $\pi$, range $[-3, 3]$
Pause — copy the key points for $y = 2\sin x$ and the parameters for $y = 3\cos(2x)$ (amplitude 3, period $\pi$, range $[-3,3]$) into your book.
Fill in the blank: For $y = 4\sin x - 1$, the range is $[\,\underline{\hspace{40px}},\ \underline{\hspace{40px}}\,]$.
We just saw worked examples for amplitude, period, and vertical shift. That raises a question: what mistakes do students make when reading off period or amplitude from the general form under exam conditions? This card answers it → Trap 1: larger $b$ means shorter period (not longer); Trap 2: amplitude is always $|a|$, never negative.
Larger $b$ → shorter period (compression); smaller $b$ → longer period (stretch); Amplitude = $|a|$, always positive — never write a negative amplitude
Pause — copy the two period/amplitude traps: larger $b$ compresses the period; amplitude $= |a|$ is always positive (never write $a = -2$ means amplitude $-2$) into your book.
Odd one out: Which of these does NOT change the amplitude of $y = \sin x$?
Quick-fire practice · 5 reps
$y = \sin x$ — state amplitude and period.
$y = 2\cos x$ — state amplitude and period.
$y = \sin(2x)$ — state amplitude and period.
$y = \cos x + 2$ — state amplitude, period, and range.
Sketch $y = 3\sin x$ for $0 \leq x \leq 2\pi$ and describe the key features.
Teach it back: Explain in 1–2 sentences why increasing $b$ in $y = \sin(bx)$ shortens the period rather than lengthening it.
The points $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, $(2\pi, 0)$ form a smooth wave — the sine curve. The cosine curve uses $(0, 1)$, $(\frac{\pi}{2}, 0)$, $(\pi, -1)$, $(\frac{3\pi}{2}, 0)$, $(2\pi, 1)$, giving the same wave shape shifted left by $\frac{\pi}{2}$.
Return to your original answer from Section 01. What did you get right? What has changed in your thinking?
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. 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)
View comprehensive answer
Amplitude = 3 [0.5], Period = $\pi$ [0.5].
Max points at $(0, 3)$, $(\pi, 3)$, $(2\pi, 3)$ [1.5].
Min points at $(\frac{\pi}{2}, -3)$, $(\frac{3\pi}{2}, -3)$ [1.5].
Q2. Find the exact period and range of $y = 2\sin\left(\frac{x}{2}\right) + 1$. (3 marks)
View comprehensive answer
Period = $\frac{2\pi}{\frac{1}{2}} = 4\pi$ [1].
Amplitude = 2 [0.5].
Range = $[1 - 2, \, 1 + 2] = [-1, \, 3]$ [1.5].
Q3. Explain why $y = \cos x$ can be obtained from $y = \sin x$ by a horizontal translation. State the exact size and direction of this translation. (3 marks)
View comprehensive answer
$\cos x = \sin\left(x + \frac{\pi}{2}\right)$ [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].
Comprehensive answers (click to reveal)
Drill 1: Amplitude $1$, Period $2\pi$ · 2: Amplitude $2$, Period $2\pi$ · 3: Amplitude $1$, Period $\pi$ · 4: Amplitude $1$, Period $2\pi$, Range $[1, 3]$ · 5: Amplitude $3$, Period $2\pi$, Range $[-3, 3]$
Q1 (4 marks): Amplitude = 3 [0.5], Period = $\pi$ [0.5], max at $(0,3)$, $(\pi,3)$, $(2\pi,3)$ [1.5], min at $(\frac{\pi}{2},-3)$, $(\frac{3\pi}{2},-3)$ [1.5].
Q2 (3 marks): Period = $4\pi$ [1], Amplitude = 2 [0.5], Range = $[-1, 3]$ [1.5].
Q3 (3 marks): $\cos x = \sin(x + \frac{\pi}{2})$ [1]. Replacing $x$ with $x + \frac{\pi}{2}$ [1]. Translation $\frac{\pi}{2}$ to the left [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering sine and cosine graph questions. Lighter alternative to the boss.
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