Definition D1
For an acute angle θ and a right triangle T having θ as one acute angle, s(θ, T) is the length of the side opposite θ in T divided by the length of the hypotenuse.
Proof.
(1)A right triangle having θ as one acute angle exists.A3
(2)In that triangle, the side opposite the right angle is called the hypotenuse, and the side opposite θ the height.convention
(3)The length of the hypotenuse is nonzero, so the division is defined.A4
(4)Set s(θ, T) = height ÷ hypotenuse. This value still depends on T.(2), (3), convention
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Lemma L1
Two right triangles having the same acute angle θ are similar.
Proof.
(1)Let T = △ABC have a right angle at C with ∠A = θ, and T′ = △A′B′C′ a right angle at C′ with ∠A′ = θ.assumption
(2)∠C = ∠C′ = 90°(1)
(3)∠A = ∠A′ = θ(1)
(4)Two pairs of angles are equal, so the triangles are similar under the correspondence A↔A′, B↔B′, C↔C′.(2), (3), A1
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Lemma L2
If △ABC and △A′B′C′ are similar under A↔A′, B↔B′, C↔C′, then BC/AB = B′C′/A′B′
Proof.
(1)Set a = BC, c = AB, a′ = B′C′, c′ = A′B′.convention
(2)a, c, a′, c′ are all greater than 0.A4
(3)BC and B′C′, AB and A′B′ are corresponding sides, so a/a′ = c/c′assumption, A2, (2)
(4)a c′ = a′ c(3), A5, (2)
(5)Dividing both sides by c c′ gives (a c′)/(c c′) = (a′ c)/(c c′)(4), (2), E
(6)c c′ = c′ c, so a/c = a′/c′(5), A7, A6
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Theorem T1
For any two right triangles T, T′ having the same acute angle θ, s(θ, T) = s(θ, T′)
Proof.
(1)Setting T = △ABC, T′ = △A′B′C′ as in L1, the two triangles are similar.L1
(2)∠C is the right angle, so the hypotenuse is AB; ∠A = θ, so the side opposite θ is BC.(1), D1
(3)In T′ likewise, the hypotenuse is A′B′ and the side opposite θ is B′C′.(1), D1
(4)BC/AB = B′C′/A′B′(1), L2
(5)s(θ, T) = BC/AB, s(θ, T′) = B′C′/A′B′(2), (3), D1
(6)s(θ, T) = s(θ, T′)(4), (5)
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Corollary C
sin θ is defined as a function of the angle θ alone.
Proof.
(1)A right triangle T with angle θ exists.A3
(2)The value of s(θ, T) is independent of the choice of T.T1
(3)Therefore setting sin θ = s(θ, T) assigns a single value to each angle θ.(1), (2), D1
(4)adjacent ÷ hypotenuse and opposite ÷ adjacent are likewise independent of T by the same argument, and are defined as cos θ and tan θ respectively.L1, L2, A3, A4
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