Chapter review

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

1 radian ≈ 57° 16′ with π = 22/7.

l = rθ

θ in radians.

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

Each wherever its terms are defined.

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

Cos is even; sin and tan are odd.

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

n ∈ Z.

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

Opposite sign in the middle.

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

Same sign in the middle.

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.

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

A, B, A ± B not odd multiples of π/2.

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

A, B, A ± B not multiples of π.

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).

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.

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.

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).