Trigonometric Functions
Trigonometric functions of any angle
State the sign, domain and range of each trigonometric function.
To define trigonometric functions for any angle, use the unit circle (radius 1, centre the origin). Turn a radius through an angle x from the positive x-axis; it ends at a point P(a, b). Then cos x = a and sin x = b: the point is always (cos x, sin x).
Since a² + b² = 1, we get sin²x + cos²x = 1. Dividing by cos²x gives 1 + tan²x = sec²x; dividing by sin²x gives 1 + cot²x = cosec²x. The other functions are tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x and cosec x = 1/sin x.
Reading the points where the circle meets the axes:
| Angle | 0 | π/2 | π | 3π/2 | 2π |
|---|---|---|---|---|---|
| Point | (1, 0) | (0, 1) | (–1, 0) | (0, –1) | (1, 0) |
| cos | 1 | 0 | –1 | 0 | 1 |
| sin | 0 | 1 | 0 | –1 | 0 |
- sin x = 0 on the x-axis, at x = nπ (n an integer).
- cos x = 0 on the y-axis, at x = (2n + 1)π/2, the odd multiples of π/2.
- After a full turn the point comes back, so sin and cos repeat every 2π. tan and cot repeat sooner, every π.
The signs follow from the signs of a and b in each quadrant. A common memory aid is 'All Silver Tea Cups', going anticlockwise from the first quadrant:
| Quadrant | I | II | III | IV |
|---|---|---|---|---|
| Positive | all six | sin, cosec | tan, cot | cos, sec |
So sin 250° is negative (quadrant III), cos 300° is positive (quadrant IV), and tan 100° is negative (quadrant II).
For negative angles: sin(–x) = –sin x and cos(–x) = cos x. The minus sign comes out of sin, tan, cot and cosec, but cos and sec simply drop it. So sin(–π/2) = –1 and cos(–π/3) = 1/2.
Domains and ranges, which the graphs show clearly:
| Function | Domain | Range | Period |
|---|---|---|---|
| sin x, cos x | R | [–1, 1] | 2π |
| tan x | R except odd multiples of π/2 | R | π |
| cot x | R except multiples of π | R | π |
| sec x | R except odd multiples of π/2 | (–∞, –1] ∪ [1, ∞) | 2π |
| cosec x | R except multiples of π | (–∞, –1] ∪ [1, ∞) | 2π |
The graph of sin x is a wave between –1 and 1, crossing zero at every multiple of π; cos x is the same wave moved left by π/2. The graph of tan x shoots up and down near vertical lines at x = ±π/2, ±3π/2, ..., where cos x = 0 and tan x is not defined; it never touches those lines.
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