From triangles to functions
In right triangles, sine, cosine and tangent are ratios for acute angles. Using the unit circle, they become trigonometric functions defined for any angle, positive or negative, even more than 360°.
For an angle θ measured from the positive x-axis, the point where the terminal arm meets the unit circle is:
(cos θ, sin θ)
and tan θ = sin θ / cos θ.
Periodic behaviour
As the angle keeps turning, the values repeat. Trig functions are periodic.
| Function | Period | Range |
|---|---|---|
| y = sin x | 360° (2π) | −1 ≤ y ≤ 1 |
| y = cos x | 360° (2π) | −1 ≤ y ≤ 1 |
| y = tan x | 180° (π) | all real numbers |
Graph of y = sin x
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| sin x | 0 | 1 | 0 | −1 | 0 |
A smooth wave starting at 0, rising to 1, back to 0, down to −1, and back to 0.
Graph of y = cos x
| x | 0° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|
| cos x | 1 | 0 | −1 | 0 | 1 |
The same wave shape, shifted 90° left of sine.
Graph of y = tan x
Tangent has vertical asymptotes where cos x = 0 (90°, 270°, …), and repeats every 180°.
Key features of sinusoidal graphs
| Feature | For y = sin x |
|---|---|
| Amplitude | 1 (half of max − min) |
| Period | 360° |
| Midline (axis) | y = 0 |
| Maximum | 1 |
| Minimum | −1 |
Sine and cosine graphs are sinusoidal: smooth repeating waves.
Signs in the quadrants (CAST)
| Quadrant | Positive |
|---|---|
| I (0°–90°) | All |
| II (90°–180°) | Sin |
| III (180°–270°) | Tan |
| IV (270°–360°) | Cos |
Memory aid: CAST, starting from quadrant IV and going counterclockwise.
Real-world periodic phenomena
- Ocean tides
- Daylight hours through the year
- Sound waves and alternating current
- A Ferris wheel rider’s height
- Seasonal temperatures
Worked examples
Example 1: sin 270°? −1.
Example 2: Period of y = cos x in radians? 2π.
Example 3: In which quadrants is cosine negative? II and III.
Common mistakes
- Thinking sin and cos can exceed 1.
- Forgetting tan’s asymptotes.
- Mixing degrees and radians.
Try it yourself
- cos 180°?
- sin 450°? (Hint: subtract 360°.)
- Amplitude and period of y = sin x?
- Where does y = tan x have asymptotes between 0° and 360°?
Answers: 1) −1 2) 1 3) 1 and 360° 4) 90° and 270°
Quick recap
- Trigonometric functions extend sin, cos and tan using the unit circle.
- They are periodic: sin and cos have period 360°; tan has period 180°.
- Sine and cosine graphs are sinusoidal with amplitude 1.