Lesson 13 of 13 · 17 min

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Must-know facts

18 facts

1π radians = 180°, so 1° = π/180 radian and 1 radian = 180°/π.
2With π = 22/7, 1 radian ≈ 57° 16′ and 1° ≈ 0.01746 radian.
3Arc length l = rθ, with θ in radians.
41° = 60′ and 1′ = 60″.
5On the unit circle, the point at angle x is (cos x, sin x), so cos² x + sin² x = 1.
61 + tan² x = sec² x and 1 + cot² x = cosec² x.
7sin x = 0 ⇔ x = nπ; cos x = 0 ⇔ x = (2n + 1)π/2, n ∈ Z.
8tan and sec are undefined at odd multiples of π/2; cot and cosec at multiples of π.
9sin and cos have range [−1, 1]; sec and cosec have range (−∞, −1] ∪ [1, ∞); tan and cot have range R.
10Periods: 2π for sin, cos, sec, cosec; π for tan and cot.
11cos (−x) = cos x, sin (−x) = −sin x.
12Positive in each quadrant: all (I), sin and cosec (II), tan and cot (III), cos and sec (IV).
13cos (A ± B) = cos A cos B ∓ sin A sin B; sin (A ± B) = sin A cos B ± cos A sin B.
14tan (A + B) = (tan A + tan B)/(1 − tan A tan B).
15cos 2x = cos² x − sin² x = 2 cos² x − 1 = 1 − 2 sin² x; sin 2x = 2 sin x cos x.
16sin 3x = 3 sin x − 4 sin³ x; cos 3x = 4 cos³ x − 3 cos x.
17sin 15° = (√3 − 1)/(2√2), cos 15° = (√3 + 1)/(2√2), tan 15° = 2 − √3 and tan 75° = 2 + √3.
18sin A + sin B = 2 sin ((A + B)/2) cos ((A − B)/2).

Common traps

Where marks are lost

Using l = rθ with θ in degrees.

Convert first: θ must be in radians. For r = 21 cm and 60°, θ = π/3 and l = 7π = 22 cm (π = 22/7), not 21 × 60 cm.

Giving the sign of a value from the acute angle alone, e.g. cos 120° = 1/2.

Find the quadrant first. 120° is in the second quadrant, where cos is negative, so cos 120° = −1/2.

Writing sin (A + B) = sin A + sin B or cos 2x = 2 cos x.

Trigonometric functions are not linear. Use sin (A + B) = sin A cos B + cos A sin B and cos 2x = 2 cos² x − 1; test with A = B = π/4 if in doubt.

Mixing up the middle sign of the cosine formulas.

cos (A + B) has a minus in the middle and cos (A − B) a plus; for sine the middle sign matches the bracket.

Taking cos (−x) = −cos x.

Cosine is even: cos (−x) = cos x. Sine and tangent are odd: sin (−x) = −sin x, tan (−x) = −tan x.

Stating the period of tan x as 2π.

tan (π + x) = tan x, so tan and cot repeat every π; sin, cos, sec and cosec repeat every 2π.

Giving the range of sec x as [−1, 1] or R.

|sec x| = 1/|cos x| ≥ 1, so the range is (−∞, −1] ∪ [1, ∞); the same holds for cosec x.

Swapping sin and cos in π ± x or 2π − x allied angles.

Names change only with π/2 or 3π/2. With π and 2π the function keeps its name; only the sign needs thought.

Forgetting the conditions on tan (A + B) and tan 2x.

tan (A + B) needs A, B and A + B away from odd multiples of π/2; tan 2x = 2 tan x/(1 − tan² x) fails where tan² x = 1.

Formulas

14 to know

Degrees and radians

π radians = 180°; radian measure = (π/180) × degree measure

1 radian ≈ 57° 16′ with π = 22/7.

Arc length

l = rθ

θ in radians.

Pythagorean identities

sin² x + cos² x = 1; 1 + tan² x = sec² x; 1 + cot² x = cosec² x

Each wherever its terms are defined.

Negative angles

sin (−x) = −sin x; cos (−x) = cos x; tan (−x) = −tan x

Cos is even; sin and tan are odd.

Periodicity

sin (2nπ + x) = sin x; cos (2nπ + x) = cos x; tan (nπ + x) = tan x

n ∈ Z.

Cosine of sum and difference

cos (A ± B) = cos A cos B ∓ sin A sin B

Opposite sign in the middle.

Sine of sum and difference

sin (A ± B) = sin A cos B ± cos A sin B

Same sign in the middle.

Allied angles

sin (π/2 − θ) = cos θ; cos (π − θ) = −cos θ; sin (π + θ) = −sin θ; cos (2π − θ) = cos θ

π/2 swaps sin and cos; π and 2π keep the name.

Tangent of sum and difference

tan (A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

A, B, A ± B not odd multiples of π/2.

Cotangent of sum and difference

cot (A ± B) = (cot A cot B ∓ 1)/(cot B ± cot A)

A, B, A ± B not multiples of π.

Double angle

sin 2x = 2 sin x cos x; cos 2x = cos² x − sin² x = 2 cos² x − 1 = 1 − 2 sin² x; tan 2x = 2 tan x/(1 − tan² x)

Also sin 2x = 2 tan x/(1 + tan² x), cos 2x = (1 − tan² x)/(1 + tan² x).

Triple angle

sin 3x = 3 sin x − 4 sin³ x; cos 3x = 4 cos³ x − 3 cos x; tan 3x = (3 tan x − tan³ x)/(1 − 3 tan² x)

From 3x = 2x + x.

Sum to product

sin A ± sin B = 2 sin ((A ± B)/2) cos ((A ∓ B)/2); cos A + cos B = 2 cos ((A + B)/2) cos ((A − B)/2); cos A − cos B = −2 sin ((A + B)/2) sin ((A − B)/2)

Mind the minus sign in cos A − cos B.

Product to sum

2 sin A cos B = sin (A + B) + sin (A − B); 2 cos A cos B = cos (A + B) + cos (A − B); −2 sin A sin B = cos (A + B) − cos (A − B)

Also 2 cos A sin B = sin (A + B) − sin (A − B).

Key terms

14 terms

Initial side: The position of the ray before it turns.
Terminal side: The position of the ray after it has turned through the angle.
Vertex: The fixed point about which the ray turns.
Positive angle: An angle traced by an anticlockwise turn; a clockwise turn gives a negative angle.
Degree: 1/360 of a full revolution; divided into 60 minutes, each of 60 seconds.
Radian: The angle at the centre made by an arc equal in length to the radius.
Unit circle: The circle of radius 1 centred at the origin, on which cos x and sin x are the coordinates.
Quadrantal angle: An integer multiple of π/2, whose terminal side lies along an axis.
Period: The smallest positive shift after which a function's values repeat: 2π for sin and cos, π for tan.
Allied angles: Angles such as π/2 ± x, π ± x and 2π − x, whose trigonometric values are tied to those of x.
Compound angle: A sum or difference such as x + y or x − y of two angles.
Identity: An equation true for every value of the variables where both sides are defined.
Even function: A function with f(−x) = f(x), such as cos x.
Odd function: A function with f(−x) = −f(x), such as sin x and tan x.