Extended. Sine, cosine and tangent are not just for acute angles in triangles. Let the angle keep turning past 90° and their values trace out waves that repeat for ever.
\(y = \sin x\) and \(y = \cos x\) are waves between \(-1\) and \(1\) that repeat every \(360^\circ\). \(y = \tan x\) repeats every \(180^\circ\) and has vertical asymptotes at \(90^\circ\) and \(270^\circ\). Their symmetry gives a second angle with the same value: \(\sin(180^\circ - x) = \sin x\), \(\cos(360^\circ - x) = \cos x\), \(\tan(180^\circ + x) = \tan x\).
Fig. 1: y = sin x (solid) and y = cos x (dashed) for 0° to 360°, plotted from computed values. The graphs cross where sin x = cos x, at 45° and 225°
Section 01
Sine and cosine waves.
In the lab, send the point round the unit circle and watch its height draw the sine graph and its horizontal position draw the cosine graph. Pause at \(90^\circ\), \(180^\circ\) and \(270^\circ\) and read off the values.
Feature
\(y = \sin x\)
\(y = \cos x\)
Value at \(0^\circ\)
\(0\)
\(1\)
Maximum \(1\) at
\(90^\circ\)
\(0^\circ\) and \(360^\circ\)
Minimum \(-1\) at
\(270^\circ\)
\(180^\circ\)
Zeros (crosses the \(x\)-axis)
\(0^\circ\), \(180^\circ\), \(360^\circ\)
\(90^\circ\), \(270^\circ\)
Period
\(360^\circ\)
\(360^\circ\)
Line of symmetry
\(x = 90^\circ\) (and \(x = 270^\circ\))
\(x = 180^\circ\)
Worked example A table of values
Complete a table of values of \(y = \cos x\) for \(x = 0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ\), and describe the shape.
The values fall from 1 to \(-1\) and rise back to 1, symmetrical about \(x = 180^\circ\).
Answer: \(1,\ 0.5,\ -0.5,\ -1,\ -0.5,\ 0.5,\ 1\): a smooth U-shaped wave (a curve, never straight lines between the points).
Worked example Where the graphs cross
Use Fig. 1 to find the values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\sin x = \cos x\).
The graphs cross twice: once while both are positive and once while both are negative.
At \(45^\circ\), \(\sin 45^\circ = \cos 45^\circ = \dfrac{\sqrt{2}}{2}\).
At \(225^\circ\), both equal \(-\dfrac{\sqrt{2}}{2}\).
Answer: \(x = 45^\circ\) and \(x = 225^\circ\)
Examiner trap
Sketching sine and cosine with sharp corners at the top and bottom, or as straight zig-zags. The graphs are smooth curves that turn gently at their maximum and minimum.
Section 02
The tangent graph.
\(\tan x = \dfrac{\sin x}{\cos x}\), so wherever \(\cos x = 0\) the tangent is undefined. The graph shoots off to infinity there, leaving vertical asymptotes.
Zeros: \(\tan x = 0\) at \(0^\circ\), \(180^\circ\) and \(360^\circ\).
Asymptotes: vertical dashed lines at \(x = 90^\circ\) and \(x = 270^\circ\). The curve gets ever closer but never touches them.
Period \(180^\circ\): the branch from \(90^\circ\) to \(270^\circ\) is an exact copy of the branch from \(-90^\circ\) to \(90^\circ\).
No maximum or minimum: \(\tan x\) takes every value, so \(\tan x = 5\) has solutions while \(\sin x = 5\) has none.
Given that \(\tan 35^\circ = 0.700\), find another angle between \(0^\circ\) and \(360^\circ\) whose tangent is \(0.700\).
The tangent graph repeats every \(180^\circ\).
\(35^\circ + 180^\circ = 215^\circ\).
Answer: \(x = 215^\circ\)
Worked example Counting solutions
How many solutions does each equation have for \(0^\circ \le x \le 360^\circ\)? (a) \(\sin x = 0.4\) (b) \(\tan x = 5\) (c) \(\cos x = -1\)
(a) The line \(y = 0.4\) cuts the sine wave twice (once on each side of \(x = 90^\circ\)).
(b) The line \(y = 5\) cuts each branch of the tangent graph once: between \(0^\circ\) and \(90^\circ\), and between \(180^\circ\) and \(270^\circ\).
(c) \(y = -1\) only touches the cosine graph at its minimum, \(x = 180^\circ\).
Answer: (a) 2 (b) 2 (c) 1
Examiner trap
Drawing the tangent curve across the asymptote at \(90^\circ\). Each branch climbs up towards \(+\infty\) on the left of the asymptote and comes back from \(-\infty\) on the right.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
For \(0^\circ \le x \le 360^\circ\), where does \(y = \cos x\) take its maximum value?
Quick check+10 XP
What is the period of \(y = \tan x\)?
Quick check+10 XP
Which point lies on the graph of \(y = \sin x\)?
Quick check+10 XP
How many solutions does \(\sin x = 0.5\) have for \(0^\circ \le x \le 360^\circ\)?
Section 03
Symmetry gives the second angle.
A calculator gives only one angle for each value. The symmetry of the graph gives the rest. Sketch the graph, draw the horizontal line, and read across.
If the calculator gives \(\alpha\) for
the other angle in \(0^\circ\) to \(360^\circ\) is
why
\(\sin x\)
\(180^\circ - \alpha\)
symmetry about \(x = 90^\circ\)
\(\cos x\)
\(360^\circ - \alpha\)
symmetry about \(x = 180^\circ\)
\(\tan x\)
\(180^\circ + \alpha\)
period \(180^\circ\)
Worked example Same sine
Given that \(\sin 40^\circ = 0.643\), find the other angle between \(0^\circ\) and \(360^\circ\) with \(\sin x = 0.643\).
The sine graph is symmetrical about \(x = 90^\circ\).
\(180^\circ - 40^\circ = 140^\circ\).
Answer: \(x = 140^\circ\)
Worked example Negative cosine
Given that \(\cos 70^\circ = 0.342\), find all the angles between \(0^\circ\) and \(360^\circ\) with \(\cos x = -0.342\).
Cosine is negative between \(90^\circ\) and \(270^\circ\), so both answers lie there.
The cosine graph has rotational symmetry about the point \((90^\circ, 0)\), so \(\cos(180^\circ - 70^\circ) = -\cos 70^\circ\), giving \(x = 110^\circ\).
Symmetry about \(x = 180^\circ\): \(360^\circ - 110^\circ = 250^\circ\).
Answer: \(x = 110^\circ\) and \(x = 250^\circ\)
Your turn
Given that \(\sin 38^\circ = 0.616\), find the two angles between \(0^\circ\) and \(360^\circ\) with \(\sin x = -0.616\).
Sine is negative between \(180^\circ\) and \(360^\circ\). The two angles are \(180^\circ + 38^\circ = 218^\circ\) and \(360^\circ - 38^\circ = 322^\circ\).
Examiner trap
Using the sine pattern \(180^\circ - \alpha\) for cosine. For cosine the partner of \(70^\circ\) is \(290^\circ\) (\(360^\circ - 70^\circ\)), not \(110^\circ\). Always check with a quick sketch.
Stage 2 · Exam
Exam-style questions
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Finish the four checks above to unlock the exam questions
Exam style+20 XP
Given that \(\sin 25^\circ = 0.4226\), find the other value of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\sin x = 0.4226\).
Exam style+20 XP
Given that \(\cos 50^\circ = 0.643\), find all values of \(x\) between \(0^\circ\) and \(360^\circ\) for which \(\cos x = -0.643\).
Exam style+20 XP
For \(0^\circ \le x \le 360^\circ\), for which values of \(x\) is \(\tan x\) undefined?
Exam style+20 XP
For \(0^\circ \le x \le 360^\circ\), the graphs of \(y = \sin x\) and \(y = \cos x\) intersect at
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