Graphs of Sine, Cosine and Tangent
The graphs of sine, cosine and tangent are the wave shapes behind every Edexcel IGCSE Maths question that asks for more than one angle. On this page you will learn the shape and key values of y = sin x, y = cos x and y = tan x, why the tangent curve races off to its asymptotes at 90 and 270 degrees, and how the symmetry of each curve gives you every solution in a given range instead of just the one your calculator shows. Work through the plotted examples, then scroll down to the auto-marked practice questions and read the answers straight off the curves.
Where the graphs of sine, cosine and tangent come from
Take a circle of radius 1 with its centre at the origin, and let a point P travel anticlockwise around it. Measure the angle \(x\) from the positive horizontal axis. Then P has coordinates \((\cos x,\ \sin x)\): the across step is the cosine and the up step is the sine.
As P goes all the way round, the up step rises to 1, falls back through 0 to \(-1\), and returns to 0. Plotting that height against the angle draws the sine curve. Plotting the across step draws the cosine curve. Dividing one by the other, \(\tan x=\dfrac{\sin x}{\cos x}\), draws the tangent curve, which breaks wherever the across step is zero because you cannot divide by 0.
Turning a table into a curve
- Set the calculator to degrees and work out the value every 30 degrees from 0 to 360.
- Plot the points, remembering the exact zeros and turning points rather than rounding them.
- Join them with a smooth curve, never a series of straight lines.
- For the tangent curve, draw the asymptotes at 90 and 270 degrees first, then let each branch climb towards them without ever touching.
The shape of each curve
y = sin x
Starts at 0, maximum 1 at 90 degrees, minimum \(-1\) at 270 degrees, and repeats every 360 degrees.
y = cos x
The same wave shifted 90 degrees: starts at 1, minimum \(-1\) at 180 degrees, and repeats every 360 degrees.
y = tan x
Zero at 0, 180 and 360 degrees, with asymptotes at 90 and 270 degrees. It repeats every 180 degrees and has no maximum or minimum.
Key values to know by heart
| Angle | 0 | 90 | 180 | 270 | 360 |
|---|---|---|---|---|---|
| sin | 0 | 1 | 0 | −1 | 0 |
| cos | 1 | 0 | −1 | 0 | 1 |
| tan | 0 | undefined | 0 | undefined | 0 |
Those five columns are enough to sketch any of the three curves quickly in an exam, and they are the values a question will ask you to read off.
Call the angle your calculator gives you \(x_1\). It can be negative, and that is fine. The second angle is \(180^\circ - x_1\) for sine, \(-x_1\) for cosine, and \(x_1 + 180^\circ\) for tangent. Then add or subtract \(360^\circ\) as many times as you need to move each answer into the range the question asked for.
That last step is the one students skip. For a negative value the calculator hands back a negative angle, so both answers need shifting before they are worth writing down.
Worked examples
Example 1: reading key values
The graph of \(y=\sin x\) is drawn for \(0^\circ \le x \le 360^\circ\). Write down the coordinates of the maximum point, and every value of \(x\) where the curve crosses the horizontal axis.
What's happening?
The sine wave rises from the origin to its single peak a quarter of the way along, so the maximum sits at 90 degrees and its height is 1.
It meets the horizontal axis at the start, the halfway point and the end, which is why three values are wanted, not one.
Example 2: all solutions from one
Solve \(\sin x = 0.5\) for \(0^\circ \le x \le 360^\circ\).
What's happening?
The calculator hands you one angle only. Draw the line \(y=0.5\) across the sine curve and you can see it cuts the wave twice in this range.
The curve is symmetrical about \(x=90^\circ\), so the second crossing is the same distance the other side of 90 degrees, which is \(180^\circ - 30^\circ\).
Example 3: the tangent curve
For \(0^\circ \le x \le 360^\circ\), write down the equations of the asymptotes of \(y=\tan x\), then solve \(\tan x = 1\).
What's happening?
Because \(\tan x=\dfrac{\sin x}{\cos x}\), the curve breaks exactly where the cosine is zero. So the asymptotes are the lines \(x=90^\circ\) and \(x=270^\circ\).
The tangent curve repeats every 180 degrees rather than every 360, so the second solution is one full pattern further on, not a reflection.
Example 4: beyond 360 degrees
Given that \(\sin 25^\circ = k\), find all the solutions of \(\sin x = k\) in the range \(360^\circ \le x \le 720^\circ\).
What's happening?
The wave keeps going in both directions and repeats exactly every 360 degrees, so every solution you find in the first cycle gives you another one a full period later.
The first three lines find the pair in the first 360 degrees; the last four shift that pair one whole period along. Extending the sketch past 360 degrees makes it obvious that two answers are needed here.
Example 5: when the value is negative
Solve \(\sin x = -0.5\) for \(0^\circ \le x \le 360^\circ\).
What's happening?
The calculator gives \(-30^\circ\), which is outside the range asked for. Do not discard it: it is a genuine solution, just drawn one whole turn back.
Use it to get the partner, \(180^\circ - (-30^\circ) = 210^\circ\), then bring the calculator angle itself into range by adding \(360^\circ\), giving \(330^\circ\). So the two answers are \(210^\circ\) and \(330^\circ\), both below the axis on the sine curve, which is exactly where a negative value should put them.
Key points
- Sine and cosine never leave the range \(-1\) to 1, so an equation such as \(\sin x = 2\) has no solutions at all.
- Sine and cosine repeat every 360 degrees; tangent repeats every 180 degrees.
- The cosine curve is the sine curve shifted 90 degrees to the left, which is why it starts at its maximum.
- Between 0 and 360 degrees the tangent curve has asymptotes at 90 degrees and 270 degrees, and never touches them. Past that range they repeat every 180 degrees.
- Use \(180^\circ - x_1\) for sine, \(-x_1\) for cosine and \(x_1 + 180^\circ\) for tangent to get the partner, then shift by \(360^\circ\) into the range you were asked for.
- A quick sketch with the range marked on it shows how many answers the question wants.
Common pitfalls
- Giving only the calculator answer. The inverse key returns one angle, and for a negative value that angle is usually negative and outside the range; the graph is what supplies the rest.
- Using the sine symmetry rule \(180^\circ - x\) on a cosine question. Check which curve you are on before reflecting.
- Leaving the calculator in radian mode, which turns every answer into a decimal that looks nothing like a whole number of degrees.
- Drawing the tangent curve as a wave. It has no maximum and no minimum, and each branch runs off to the asymptote.
- Forgetting the endpoints. In \(0^\circ \le x \le 360^\circ\) both 0 and 360 are inside the range, so \(\sin x = 0\) has three solutions.
- Rounding the key values. Write \(\sin 90^\circ = 1\) exactly, not 0.99 or 1.0 from a mis-set calculator.