In today's blog, I will talk about some of the properties of equilateral triangles.
I will first show that a 60 °-60 ° - 60 ° triangle is an equilateral triangle. Then, I will show that properties of this triangle establish that:
cos 60° = sin 30 ° = 1/2
sin 60° = cos 30 ° = √3/2
First, a definition:
Definition 1: Equilateral Triangle
An equilateral triangle is a triangle where all sides are equal.
Lemma 1: A 60 °-60 ° - 60 ° triangle is an equilateral triangle.
Proof:
AB ≅ AC ≅ BC since congruent angles imply congruent sides (see Corollary 1, here)
QED
Corollary 1.1: Trigonometric Properties
sin 30 ° = 1/2
cos 30 ° = √3/2
Proof:
(1) Let ABC be an 60 ° - 60 ° - 60 ° triangle.
(2) From Lemma 1 above, ABC is an equilateral triangle.
(3) Let AD be a line that bisects ∠ BAC so that ∠ DAC is 30 °
(4) We know that ∠ ADC is a right angle and BD ≅ CD since:
(a) triangle BAD ≅ triangle CAD by Side-Angle-Side (see Postulate 1, here)
(b) From congruent triangles (see Definition 1, here), BD ≅ CD and ∠ ADC ≅ ∠ ADB
(c) Since ∠ ADC and ∠ ADB add up to 180 ° (see Postulate 1, here), they must be right angles.
(5) Assume that AC = 1
(6) Then sin 30 ° = CD/AC = (1/2)/1 = (1/2).
(7) Using the Pythagorean Theorem (see Theorem 1, here):
AD = √(AC)2 - (CD)2 = √(1)2 - (1/2)2 =
= √3/4 = √3/2
(8) cos 30 ° = AD/AC = AD/1 = √3/4 = √3/2
QED
Corollary 1.2
cos 60 ° = 1/2
sin 60 ° = √3/2
Proof:
(1) sin(30°) = 1/2 [Corollary 1.1 above]
(2) cos(60°)=cos(90° - 30°) = sin(30 °) [See Corollary 1.7, here]
(3) cos(30 °) = √3/2 [Corollary 1.1 above]
(4) sin(60°)=sin(90° - 30°) = cos(30 °) [See Corollary 1.8, here]
QED
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