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:


Cscθ (Cosecant) = Hypotenuse over Opposite

Secθ (Secant) = Hypotenuse over Adjacent

Cotθ (Cotangent) = Adjacent over Opposite

Reciprical Ratios: Cosecant Secant Cotangent flippity flop Sine Cosine Tangent: Trig Math Help

Evaluating Ratios of Special Angles

Special Trigonomic ratios come from two "special" triangles


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.


Trigonometric ratios of angles over 90°

An angle in standard position has a vertex which lies on the origin and the initial arm on the positive x-axis


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°

Similar to trigonometric ratios of angles over 90° but applies to any angle between 0° to 360°
Trigonometric Ratios and Special Angles.avi

Trigonometric Identities

A Trigonometric identity is an equation involving trigonometric ratios that is true for all values of the variable. These trigonometric identities can be substituted for other variables in order to be able to solve the equation using like terms. Trigonometric ratios should be memorized!
Memorizing Trig Identities

The Sine Law and The Co-Sine Law

When given a non-right angled triangle, one cannot solve by using the basic trigonometric ratios of Sine, Co-sine, and Tangent. In order to solve a problem as such, you would need to use either the Sine Law or 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

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

Sine and Cosine Laws When do you use each one.mp4

3-D Trigonometry

Using the knowledge of the Trigonometric ratios above, we can apply it to 3-D shapes that contain triangular sides and solve for unknown measurements such as length and width

Practise Questions!

Review of Trigonometry:

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