Sine and cosine are waves of height 1 that repeat every \(2\pi\); tangent repeats every \(\pi\) and shoots off to infinity. Learn the three basic shapes, then stretch and shift them to sketch graphs like \(y = 3\sin 2x\) and \(y = 1 - \cos 2x\) in degrees or radians.
\(y = \sin x\) and \(y = \cos x\) have period \(2\pi\) (\(360^\circ\)) and range \(-1 \le y \le 1\); \(y = \tan x\) has period \(\pi\) (\(180^\circ\)) and asymptotes at \(x = \dfrac{\pi}{2} + n\pi\). For \(y = a\sin bx + c\): amplitude \(|a|\), period \(\dfrac{2\pi}{b}\), and the centre line moves to \(y = c\).
Fig. 1: y = sin x (solid) and y = cos x (dashed) for 0 ≤ x ≤ 2π. The cosine graph is the sine graph translated π/2 to the left
Section 01
The three basic shapes.
In the lab, drag the point round the unit circle and watch its height trace out the sine graph and its across-distance trace out the cosine graph. Keep going past one full turn: the graphs simply repeat, which is why they work for angles of any size, positive or negative.
Degrees or radians: the shapes are identical; only the labels on the \(x\)-axis change (\(\pi\) for \(180^\circ\)).
Examiner trap
The tangent graph repeats every \(\pi\), not every \(2\pi\), and it has no maximum or minimum. Draw its asymptotes as dashed vertical lines and make the branches approach them without touching.
Section 02
Stretches and translations.
Every graph in Paper 1 is one of the three basic shapes changed by the transformations you already know. Read each number in \(y = a\sin(bx) + c\) as one transformation and apply them one at a time.
\(y = a\sin x\): stretch parallel to the \(y\)-axis, factor \(a\). The amplitude becomes \(|a|\); a negative \(a\) also reflects in the \(x\)-axis.
\(y = \sin bx\): stretch parallel to the \(x\)-axis, factor \(\dfrac{1}{b}\). The period becomes \(\dfrac{2\pi}{b}\) (\(\dfrac{\pi}{b}\) for tan).
\(y = \sin x + c\): translation by \(c\) up. The graph now oscillates about \(y = c\).
\(y = \sin(x + k)\): translation by \(k\) to the LEFT.
Worked example Amplitude and period
State the amplitude and period of \(y = 3\sin 2x\), and its maximum point for \(0 \le x \le \pi\).
The 3 stretches vertically: amplitude 3, so the range is \(-3 \le y \le 3\).
The 2 squeezes horizontally: period \(\dfrac{2\pi}{2} = \pi\).
\(\sin 2x = 1\) when \(2x = \dfrac{\pi}{2}\), so \(x = \dfrac{\pi}{4}\).
Answer: Amplitude 3, period \(\pi\), maximum at \(\left(\dfrac{\pi}{4}, 3\right)\)
Worked example Sketching y = 1 - cos 2x
Sketch \(y = 1 - \cos 2x\) for \(0 \le x \le \pi\), giving the coordinates of the turning points.
Start with \(y = \cos 2x\): period \(\pi\), so one full wave between 0 and \(\pi\), starting at a maximum.
\(-\cos 2x\) reflects it in the \(x\)-axis: now it starts at a minimum of \(-1\).
Adding 1 moves it up 1: it oscillates between \(0\) and \(2\) about \(y = 1\).
Minimum 0 at \(x = 0\) and \(x = \pi\); maximum 2 where \(\cos 2x = -1\), i.e. \(2x = \pi\), \(x = \dfrac{\pi}{2}\).
Answer: A single upside-down cosine wave: minima \((0, 0)\) and \((\pi, 0)\), maximum \(\left(\dfrac{\pi}{2}, 2\right)\)
Worked example A translated tangent
For \(y = \tan\left(x + \dfrac{\pi}{4}\right)\), \(0 \le x \le \pi\), find the asymptote, the \(x\)-intercept and the \(y\)-intercept.
This is \(y = \tan x\) translated \(\dfrac{\pi}{4}\) to the left.
Asymptote where \(x + \dfrac{\pi}{4} = \dfrac{\pi}{2}\): \(x = \dfrac{\pi}{4}\).
Zero where \(x + \dfrac{\pi}{4} = \pi\): \(x = \dfrac{3\pi}{4}\).
At \(x = 0\): \(y = \tan\dfrac{\pi}{4} = 1\).
Answer: Asymptote \(x = \dfrac{\pi}{4}\); crosses the \(x\)-axis at \(\left(\dfrac{3\pi}{4}, 0\right)\) and the \(y\)-axis at \((0, 1)\)
Your turn
State the range and period of \(y = 2 - 4\cos 3x\).
\(4\cos 3x\) runs from \(-4\) to \(4\), so \(2 - 4\cos 3x\) runs from \(-2\) to \(6\): range \(-2 \le y \le 6\). Period \(\dfrac{2\pi}{3}\).
Examiner trap
\(y = \sin(x + k)\) moves the graph LEFT by \(k\), not right; and in \(y = \sin 2x\) the period halves, it does not double.
Stage 1 · Learn
Check your understanding
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What is the period of \(y = \tan x\)?
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What is the range of \(y = 4\cos x - 1\)?
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What is the period of \(y = \sin 3x\)?
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The graph of \(y = \cos x\) is translated to give \(y = \cos\left(x - \dfrac{\pi}{2}\right)\). Which graph is this?
Section 03
Reading the graph.
Questions often give a sketch and ask you to find the constants, or ask how many solutions an equation has. Use the maximum and minimum to find the vertical constants, and count crossings of a horizontal line to count solutions.
Worked example Finding the constants
The graph of \(y = a + b\cos x\), where \(b > 0\), has a maximum value of 5 and a minimum value of \(-1\). Find \(a\) and \(b\).
The centre line is halfway between max and min: \(a = \dfrac{5 + (-1)}{2} = 2\).
The amplitude is half the distance from max to min: \(b = \dfrac{5 - (-1)}{2} = 3\).
Check: \(2 + 3 = 5\) and \(2 - 3 = -1\).
Answer: \(a = 2\), \(b = 3\)
Worked example Counting solutions
How many solutions does \(3\sin 2x + 1 = 0\) have for \(-\pi < x < \pi\)?
Rearrange: \(\sin 2x = -\dfrac{1}{3}\).
\(y = \sin 2x\) has period \(\pi\), so between \(-\pi\) and \(\pi\) there are two complete waves.
Each complete wave crosses the line \(y = -\dfrac{1}{3}\) twice (the line is between \(-1\) and \(1\)).
Answer: 4 solutions
Your turn
Find the set of values of \(k\) for which \(3\sin x + 2 = k\) has exactly two solutions for \(0 \le x \le \pi\).
For \(0 \le x \le \pi\), \(y = 3\sin x + 2\) rises from 2 to 5 at \(x = \dfrac{\pi}{2}\) and falls back to 2. A horizontal line meets it twice when \(2 \le k < 5\) (at \(k = 5\) it touches once).
Examiner trap
Watch the ends of the interval. With \(0 \le x \le \pi\) the end points are included, so \(k = 2\) gives two solutions (\(x = 0\) and \(x = \pi\)); with \(0 < x < \pi\) it would give none.
Stage 2 · Exam
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The graph of \(y = a + b\sin x\), where \(b > 0\), has maximum value 7 and minimum value 1. Find \(a\) and \(b\).
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Find the equations of the asymptotes of \(y = \tan 2x\) for \(0 < x < \pi\).
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How many solutions does \(2\sin 2x = 1\) have for \(0 \le x \le 2\pi\)?
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Find the coordinates of the maximum point of \(y = 3\cos\left(x - \dfrac{\pi}{3}\right)\) for \(0 \le x \le 2\pi\).
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Trig graphs, sketched.
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