In these lessons, 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 without a calculator.

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## Special Angles: 30 and 60

## Special Angles: 45 and 90

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

## Videos

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.
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.

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°)

You can use the Mathway widget below to practice Trigonometry or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

Related Topics:

More lessons on Trigonometry

Worksheets on Trigonometry

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}

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°)

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°)

You can use the Mathway widget below to practice Trigonometry or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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