A1In a right triangle, a² + b² = c²
Pythagorean theorem (cited here, not proved)
A2trigonometric ratios depend on the angle alone
Proof 32
A3sin²θ + cos²θ = 1
Proof 33
A4opposite = hypotenuse × sin θ
Definition of sine in a right triangle
A5adjacent = hypotenuse × cos θ
Definition of cosine in a right triangle
A6sin(180°−θ) = sin θ
Sine of an obtuse angle (cited here, not proved)
A7cos(180°−θ) = −cos θ
Cosine of an obtuse angle (cited here, not proved)
A80 < θ < 180° ⟹ sin θ > 0
Sine is positive at interior angles of a triangle
A9sin 90° = 1, cos 90° = 0
Trigonometric ratios of the right angle
A10the foot of the perpendicular exists uniquely
Perpendicular dropped from a point off a line
A11H ∈ ray CB ⟺ C ≤ 90°
Position of the foot of the perpendicular
A12∠ACH + ∠ACB = 180°
When H lies on the opposite ray (straight angle)
A13PQ = |CP − CQ|
P, Q on the same ray from C
A14PQ = CP + CQ
P, Q on opposite rays from C
A15|t|² = t²
Square of the absolute value
A16(u−v)² = u²−2uv+v²
Product formula (Proof 23)
A17a(b + c) = ab + ac
Distributive law
A18a + b = b + a, ab = ba
Commutativity of addition and multiplication
A19(a+b)+c = a+(b+c)
Associativity of addition
A20(ab)c = a(bc)
Associativity of multiplication
A21a × 1 = a
Multiplicative identity
A22a+0 = a, a×0 = 0, 0² = 0
Arithmetic of 0
A23(xy)/(zy) = x/z
Cancellation of fractions (y, z ≠ 0)
A24a(−b) = −(ab)
Sign rule
Ea = b ⟹ same operation on both sides
Properties of equality
Nothing outside this list may be cited.