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In this lesson, we will learn the trigonometric functions of the special angles 30˚, 45˚, 60˚, and 90˚ and how to use them to find exact values of trigonometric expressions.

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Let us first consider 30˚ and 60˚.

These two angles form a 30˚-60˚-90˚ right triangle as shown.

The ratio of the sides of the triangle is

From the triangle we get the ratios as follows:

Combining the two tables we get:

* Example: *

Evaluate the following without using a calculator:

a) 2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

b) 3(cos 30˚)^{2} + 2 (sin 30˚)^{2}

* Solution: *

a) 2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

b) 3(cos 30˚)^{2} + 2 (sin 30˚)^{2}

This lesson shows how to find the trig ratios of the special angles and how to use them to find exact values of expressions involving sine, cosine and tangent values of 0, 30, 45, 60 and 90 degrees. This is conclusion of a two part lesson.

30-60-90 and 45-45-90 Triangles

The following video explains how the trigonometry values for the special angles are obtained.

(Errata: At time 4:10, the video should show that )

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