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Geometry

Use the properties of a right triangle to find the sine, cosine, tangent, and cotangent ratios of 30?, 45?, and 60? angles.

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After completing this tutorial, you will be able to complete the following:

- Apply the properties of a right triangle to evaluate the sine, cosine, tangent, and cotangent ratios of 30<img src../../../tutorial
s/images/teacherguid es/US820503CD_1.png\ " width=\"8\" height=\"10\" alt=\"\" border=\"0\"> and 60<img src../../../tutorial s/images/teacherguid es/US820504PU_1.png\ " width=\"8\" height=\"10\" alt=\"\" border=\"0\"> angles. - Apply the properties of a right triangle to find the sine, cosine, tangent, and cotangent ratios of a 45<img src../../../tutorial
s/images/teacherguid es/US820503CD_1.png\ " width=\"8\" height=\"10\" alt=\"\" border=\"0\"> angle.

Trigonometric ratios of a right triangle

A right triangle has one right angle and two acute angles. The side across from the right angle is called the hypotenuse and it is always the longest side in the triangle. The side across an angle (theta ) is called the opposite side. The side next to an angle is called the adjacent side. See figure below.

- The Cosine of an angle theta , denoted cos , is the ratio of the adjacent side to the hypotenuse of a right triangle.
- The Sine of an angle theta , denoted sin , is the ratio of the opposite side to the hypotenuse of a right triangle.
- The Tangent of an angle theta , denoted tan, is the ratio of the opposite side to the adjacent side in a right triangle. It is also the ratio of sin to cos.
- The Cotangent of an angle theta, denoted cot is the ratio of the opposite side to the adjacent side in a right triangle. It is also the ratio of cos to sin.

30-60-90 Triangle

When finding the trigonometric ratios of 30 and 60 angles, we use a 30-60-90 triangle. This is a special right triangle that has angle measures of 30, 60, and 90. In this Activity Object, a 30-60-90 triangle can be formed from an equilateral triangle. Recall that an equilateral triangle is a triangle with three equal sides and three angles of 60each.

The sides of the 30-60-90 triangle are as follows:

The side opposite the 30 degree angle is 1 (½ the length of the hypotenuse , see picture above). Then, using the Pythagorean Theorem, we can see that the side adjacent to the 30 degree angle is .

45-45-90 Triangle

The other special triangle used is a 45-45-90 triangle. This is a special right triangle with two angle measures of 45and an angle measure of 90. In this Activity Object, a 45-45-90 triangle can be formed from a square.

The sides adjacent to the 90 degree angle are of length 1. Then, using the Pythagorean Theorem, we obtain the length of the hypotenuse to be

There are relationships between the trigonometric ratios involving 30 and 60 angles.

- sin 30 is equal to cos 60.
- sin 60 is equal to cos 30.
- tan 30 is equal to cot 60?.
- cot 30 is equal to tan 60.

This is because the side opposite the 30 angle is the same as the side adjacent the 60 angle; thus, the ratios are the same.

There are also relationships between the trigonometric ratios involving 45 angles.

Sin 45? is equal to cos 45 . Since the 45-45-90 triangle is an isosceles triangle, the legs of the triangle are the same size. Likewise, tan 45 is equal to cot45 .

Being able to find the trigonometric ratios of these special angles is important.

When a learner is trying to find the trigonometric ratios of larger angles, such as 315, knowing how to use a right triangle to find the values of the trigonometric ratio will provide the learner with an easy way to solve more difficult trigonometric problems.

Approximate Time | 35 Minutes |

Pre-requisite Concepts | Properties of equilateral triangles, properties of squares, Pythagorean theorem, rationalizing the denominator, trigonometric ratios in the right triangle |

Course | Geometry |

Type of Tutorial | Procedural Development |

Key Vocabulary | 30-60-90 triangle, 45-45-90 triangle, cosine |