All three graphs can be rebuilt from five values at 0°, 90°, 180°, 270° and 360°. Knowing where each curve starts and where it breaks is enough to tell them apart.
This lesson follows trigonometric ratios beyond acute angles and belongs to the trigonometric ratios and graphs chapter.
What are the five key values?
Fill in this table once, and every sketch becomes a matter of joining points.
| Angle | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | −1 | 0 |
| cos x | 1 | 0 | −1 | 0 | 1 |
| tan x | 0 | undefined (asymptote) | 0 | undefined (asymptote) | 0 |
Sine and cosine stay between −1 and 1. Tangent has no upper or lower limit, and it repeats every 180°, not every 360°.
Worked example: matching a point to a graph
An original question: a curve passes through (0°, 1), (90°, 0) and (180°, −1). Which graph is it?
- The value at 0° is 1, so the curve starts at its highest point.
- Sine starts at 0, so it is not sine.
- Tangent would be 0 at 0°, so it is not tangent.
- Cosine matches all three points, so the curve is y = cos x.
Rule out the graphs that fail the first point, then confirm with a second.
The mistake that costs marks
A common slip is to treat tan 90° as a number. A student writes tan 90° = 1, or 0, because sin 90° = 1 and cos 90° = 0, and then divides carelessly.
| Step | Wrong | Right |
|---|---|---|
| Write tan 90° as a ratio | 1 ÷ 0 = 0 | sin 90° ÷ cos 90° = 1 ÷ 0, which has no value |
| Interpret it | a value of 0 | undefined, vertical break |
| On the graph | curve touches the axis | dashed vertical line at 90° |
A calculator will show an error for tan 90°. That message is the correct answer, not a fault.
Check yourself
Which of y = sin x, y = cos x and y = tan x pass through the point (180°, 0)? Which pass through (180°, −1)?
Answer
At 180°, sin 180° = 0, cos 180° = −1 and tan 180° = 0.
So sine and tangent pass through (180°, 0). Only cosine passes through (180°, −1).
What to study next
With the shapes in place, the next lesson reads angles off them. Continue with finding angles from a trigonometric graph, then practise in the trigonometry practice set. The graph evidence and fair-comparison lab offers more graph-reading practice.
For a teacher to sketch these with you, see online one-to-one Mathematics tuition.