# Trigonometry & Periodic Functions

### MCR 3U0 - Unit 5

## Evaluating Ratios of Special Angles

The First triangle comes from dividing a square with the side length of 1 diagonally to form two isosceles right-angled triangles with two 45° angles which result in the following trigonometric values:

Sin 45° = 1 over the square root of 2

Cos 45° = The Square root of 2 over 2

Tan 45° = 1

The Second triangle comes from dividing an equilateral triangle with the side length of 2 in half to form two right-angled triangles with angles 30° and 60°. These angles also have their own trigonometric values:

Sin 30° = 1 over 2

Cos 30° = The square root of 3 over 2

Tan 30° = 1 over the square root of 3

Sin 60° = The square root of 3 over 2

Cos 60° = 1 over 2

Tan 60° = The square root of 3

These special ratios are to be remembered and used as trigonometric identities to substitute into an equation to make solving easier.

## First Triangle Sin 45° Cos 45° Tan 45° | ## Second Triangle Sin 30° Cos 30° Tan 30° Sin 60° Cos 60° Tan 60° | ## Memorization Required! Make sure to memorize these just like the trigonometric identities! |

## Trigonometric ratios of angles over 90°

__Principle Angle__

The counter-clockwise angle between the initial arm and the terminal arm

__Related Acute Angle__

The angle between the terminal arm and the closest x-axis

## Trigonometric ratios of angles between 0° and 360°

## Trigonometric Identities

## The Sine Law and The Co-Sine Law

The Sine law would be used to solve the non-right angled triangle when given:

1. The measurement of two angles and the length of one side

2. The measurement of the lengths of two sides and one angle opposite to any one of these sides

The Co-Sine Law would be used to solve the non-right angled triangle when given:

1. The measurement of the lengths of two sides and the contained angle

2. The measurement of the lengths of three sides

## The Sine Law An example of a non-right angled triangle with the Sine Law equations surrounding it. | ## The Co-Sine Law An example of a non-right angled triangle with the Co-Sine Law equations displayed on the right of the diagram | ## The Co-Sine Law Continued A List of the possible Co-Sine Law equations |

## Remember to memorize the equations for Sine Law and Co-Sine Law!

## Practise Questions!

Textbook: Page 274 Questions #1-7

Primary and Reciprocal Trigonometric Ratios:

Textbook: Page 280-282 Questions #1-12,14

Trigonometric Ratios of Special Angles:

Textbook: Page 286-287 # 1-11

Angles in Standard Position, Related Angles, and CAST Rule:

Textbook: Page 292 #1-4, Page 299-301 #1-14

Trigonometric Identities:

Textbook: Page 310-311 #1-13

Sine Law and Ambiguous Case Applications:

Textbook: Page 318-320 #1-13

Cosine Law and Applications:

Textbook: Page 325-327 #1-11,14

3-D Problems:

Textbook: Page 332-333 #1-8