Measuring angles
An angle is measured by the amount of rotation of a ray about its starting point. Two units are used:
Degree: 1/360 of a full rotation.
Radian: the angle subtended at the centre of a circle by an arc equal in length to the radius.
A full circle has circumference 2πr, so a full turn is 2π radians = 360°, and
π radians = 180°, so 1 radian ≈ 57° 16′.
To convert: radians = degrees × π/180; degrees = radians × 180/π.
Degrees | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|
Radians | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
Arc length: an arc of a circle of radius r subtending θ radians at the centre has length l = rθ.
Example. A pendulum 75 cm long swings through an arc of 10 cm. The angle is θ = 10/75 = 2/15 radian.
Trigonometric functions of any angle
Take a unit circle (radius 1) centred at the origin. If a ray at angle x from the positive x-axis meets the circle at P(a, b), define
cos x = a and sin x = b.
Then tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x, cosec x = 1/sin x.
Because P lies on the unit circle, sin² x + cos² x = 1 for every x. Dividing by cos² x gives 1 + tan² x = sec² x; dividing by sin² x gives 1 + cot² x = cosec² x.
Signs in the four quadrants
Quadrant | Angle range | Positive functions |
|---|---|---|
I | 0 to π/2 | all |
II | π/2 to π | sin (and cosec) |
III | π to 3π/2 | tan (and cot) |
IV | 3π/2 to 2π | cos (and sec) |
A common memory aid is "All School To College".
Example. If cos x = −3/5 and x is in the third quadrant, then sin x = −√(1 − 9/25) = −4/5, and tan x = (−4/5)/(−3/5) = 4/3.
Domain, range and periodicity
Function | Domain | Range | Period |
|---|---|---|---|
sin x | ℝ | [−1, 1] | 2π |
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 | ℝ − (−1, 1) | 2π |
cosec x | ℝ except multiples of π | ℝ − (−1, 1) | 2π |
A function is periodic if its values repeat: sin(x + 2π) = sin x and tan(x + π) = tan x. The graph of sin x is a wave between −1 and 1, crossing the x-axis at multiples of π. The graph of cos x is the same wave shifted left by π/2.
Values at special angles and allied angles
sin(−x) = −sin x and cos(−x) = cos x (sine is odd, cosine is even).
sin(π/2 − x) = cos x and cos(π/2 − x) = sin x.
sin(π − x) = sin x and cos(π − x) = −cos x.
sin(π + x) = −sin x and cos(π + x) = −cos x.
Example. sin 31π/3 = sin(10π + π/3) = sin π/3 = √3/2, since 10π is five full turns.
Sum and difference formulas
cos(x + y) = cos x cos y − sin x sin y
cos(x − y) = cos x cos y + sin x sin y
sin(x + y) = sin x cos y + cos x sin y
sin(x − y) = sin x cos y − cos x sin y
tan(x + y) = (tan x + tan y) ⁄ (1 − tan x tan y)
Example. sin 15° = sin(45° − 30°) = (√2/2)(√3/2) − (√2/2)(1/2) = (√3 − 1)/(2√2).
Double and triple angles
sin 2x = 2 sin x cos x
cos 2x = cos² x − sin² x = 2cos² x − 1 = 1 − 2sin² x
tan 2x = 2 tan x ⁄ (1 − tan² x)
sin 3x = 3 sin x − 4 sin³ x and cos 3x = 4 cos³ x − 3 cos x
Sum-to-product formulas
sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2)
cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2)
cos x − cos y = −2 sin((x + y)/2) sin((x − y)/2)
Read backwards, they also turn products into sums: 2 sin x cos y = sin(x + y) + sin(x − y).
Summary
π radians = 180°; arc length l = rθ with θ in radians.
sin² x + cos² x = 1; signs follow "All School To College".
sin and cos have period 2π and range [−1, 1]; tan has period π.
Learn the sum, difference and double-angle formulas thoroughly; most identities follow from them.