Trigonometry & Periodic Functions
MCR 3U0 - Unit 5
Review of Trignometric Ratios (Grade 10)
Remember: SOH - CAH - TOA
Sinθ (Sine) = Opposite over Hypotenuse
Cosθ (Cosine) = Adjacent over Hypotenuse
Tanθ (Tangent) = Opposite over Adjacent
Reciprocal Ratios
As the name suggests, the reciprocal ratios are the reciprocal of the original trigonometric ratios. Therefore, the reciprocal ratios are as follows:
Secθ (Secant) = Hypotenuse over Adjacent
Cotθ (Cotangent) = Adjacent over Opposite
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
Cos 45°
Tan 45°
Second Triangle
Cos 30°
Tan 30°
Sin 60°
Cos 60°
Tan 60°
Memorization Required!
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°
Visual representation of each quadrant and it's trigonometric properties
Conditions within each quadrant
Unit Circle
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
The Co-Sine Law
The Co-Sine Law Continued
Remember to memorize the equations for Sine Law and Co-Sine Law!
3-D Trigonometry
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