In these lessons, we will learn the trigonometric ratios of the special angles 30˚, 45˚, 60˚, and 90˚ and how to use them to find exact values of trigonometric expressions without a calculator.

**Related Pages**

Lessons On Trigonometry

Inverse trigonometry

Trigonometry Worksheets

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 1:√3:2

From the triangle we get the ratios as follows:

Next, we consider the 45˚ angle that forms a 45˚-45˚-90˚ right triangle as shown. The ratio of the sides of the triangle is

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}

**How to find the trig ratios of the special angles?**

This video 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 the first part of a two part lesson. Scroll down for part 2.

**How to find the trig ratios of the special angles?**

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.

**How to use right triangle and label sides to find sin, cos, tan, cot, csc, and sec of the special angles, and of angles at multiples of 90°?**

**Examples:**

Find the exact value of each

a) cos 90°

b) tan 90°

c) sin 630°

d) cos 135°

e) tan (-405°)

f) sin 210°

g) tan (-30°)

**Easy way to find trig functions of special angles**

**Examples:**

Find the exact value of each

a) cos 300°

b) cot 180°

c) sin 1305°

d) sec (-210°)

e) csc (750°)

f) cos 270°

g) sin (-420°)

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