Day 96 – Trigonometry of 30-60-90 right triangle 1

Introduction A right triangle is a special triangle because it has one of its angle equal to 90°. However, among right triangle, there are more special triangles, these are triangles whose interior angles are commonly used in the application. These interior angles are 30°, 45°, and 60°. Of course, a right triangle cannot have all these angles, to mean, they must be two of such special features. We are going to introduce the first one. In this lesson, we are going to find the trigonometric ratios of 30-60-90 right triangle.

Vocabulary Trigonometric ratios These are ratios of three combinations of two sides of a right triangle. They are sine, cosine and tangent.

Trigonometric ratio of 30-60-90 triangle Consider an equilateral triangle of side 2 unit. When we divide it equally with a perpendicular bisector of one of the lines, we get right scalene triangle ABC. A B C K 1 2

We find the length of BC by Pythagorean theorem We find the length of BC by Pythagorean theorem. 𝐵𝐶= 2 2 − 1 2 = 4−1 = 3 Since AKC is an equilateral triangle and BC a perpendicular bisector of AK, we have the triangle of interest with interior angles as shown. 60° 30° A B C 1 2 3

Thus, we have cos 30° = 3 2 ; sin 30° = 1 2 ; tan 30° = 1 3 cos 60° = 1 2 ; sin 60° = 3 2 ; tan 60° = 3

Example 1 Find the sin 60° and tan 30° Example 1 Find the sin 60° and tan 30° . Solution We draw a 30-60-90 triangle. 1 3 2 60°

Example 1 Find the exact values of sin 60° and tan 30° Example 1 Find the exact values of sin 60° and tan 30° . Solution We draw a 30-60-90 triangle. From the triangle, we have sin 60°= 3 2 and tan 30° = 1 3 . Since the exact values are required, we don’t approximate the values to decimal places. 1 3 2 60°

homework Find the exact values of cot 60° and cos 60°.

Answers to homework cot 60° = 1 3 cos 60° = 3 2

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