Day 97 –Trigonometry of 30-60-90 right triangle 2

Introduction In the previous lessons, we dealt with trigonometric ratios of a general right triangle not considering some special cases. As discussed in the previous lesson a right triangle with complementary angles of 30 ° and 60 ° has the ratio of the lengths of its sides being 1: 3 :2. In this lesson, we will use this property to solve a triangle with complementary angles of 30 ° and 60 ° .

Vocabulary 30-60-90 right triangle This is a right triangle with complementary angles of 30 ° and 60 ° .

Consider the triangle below From the triangle above sin 30 ° = 1 2 , cos 30 ° = 3 2 and tan 30 ° = 1 3 cos 60 ° = 1 2 , sin 60 ° = 3 2 and tan 60 ° = 3 These are the trigonometric ratios that are used when solving 30-60-90 right angle. 1 𝑢𝑛𝑖𝑡 2 𝑢𝑛𝑖𝑡𝑠 3 𝑢𝑛𝑖𝑡𝑠 30 ° 60 °

The side opposite to an angle of 30 ° is half the length of the hypotenuse. Thus we find the length of the hypotenuse by multiplying the opposite side by 2. The adjacent side is 3 times the length of the opposite side. Thus if 30 ° is the reference angle, the following equations holds. Opposite side = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 2 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 side = 3 × opposite side Hypotenuse = 2 3 ×𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 side

If 60 ° is the reference angle, then the adjacent side is half the length of the hypotenuse. The opposite side is 3 times the length of the adjacent. Thus if 60 ° is the reference angle, the following equations holds. Adjacent side = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 2 Adjacent side = 3 ÷𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 side Hypotenuse = 2 3 ×𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 side

Example Find the value of a in the figure below Solution sin 60 ° = 𝑎 10 3 2 = 10 𝑎 𝑎=10× 2 3 = 20 3 = 𝟐𝟎 𝟑 𝟑 𝒊𝒏 10 𝑖𝑛 𝑎 60 °

homework Find the length of MO in the triangle below. 𝑀 𝑁 𝑂 27 𝑖𝑛 60 °

Answers to homework 27 3 2 𝑖𝑛

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