# Trigonometry

## Topics

1. Primary and Reciprocal Trigonometric Ratios
2. Evaluating Trigonometric Ratios for Special Angles
3. Exploring Trigonometric Ratios fro Angles Greater than 90
4. Evaluating Trigonometric Ratios for Any Angle Between 0 and 360
5. Trigonometric Identities
6. The Sine Law
7. The Cosine Law
8. Solving Three- Dimensional Problems by Using Trigonometry

## 1.Primary and Reciprocal Trigonometric Ratios

Key Terms:
• Reciprocal trigonometric ratio: one divided by each of the primary trigonometric ratios

- reciprocal trigonometric ratios make it easier to solve because it brings the unknown value to the numerator of the ratio

- reciprocal ratios are the opposite of the primary trig ratio (ex. sin= opposite/ hypotenuse, but csc= hypotenuse/ opposite)

- "csc" is short for cosecant, which is the opposite of sin

- "sec" is short for secant, which is the opposite of cos

- "cot" is short of cotangent, which is the opposite of tan

Primary & Reciprocal Trigonometric Ratios :

Practice Problem:
Determine the corresponding reciprocal trigonometric ratio.
1. cosA = 4/13
2. tanA = 12/25
3. sinA = 2/ 17

Solution:

*Flip the primary trigonometric ratio in order to get the reciprocal.

1. secA = 13/4
2. cotA = 25/12
3. cscA = 17/2

## 2. Evalutating Trigonometric Ratios for Special Angles

Special Triangles are used to help us find the values of the primary trigonometric ratios. The triangles can help to find the values for angles 30,45, 60.
Practice Problem:

Evaluate the following without using a calculator. Do not use decimals.
2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

Solution:
1. Use the chart to determine the values of each angle.
2. Start subbing in the values into the equation.
3. Simplify answer to lowest form.

2(1/2) + 3(1/2) -3(1)
= 2/2 + 3/2 - 3
= 1 + 3/2 -3
= 3/2 -2 [Get common denominators]
= 3/2 - 4/2
= -1/2
Trigonometry - The "Why" Behind Our "Special" Triangles - 30-60-90 and 45-45-90

## 3. Exploring Trigonometric Ratios for Angles Greater than 90

Key Terms:
• Principal Angle: the angle between the initial arm and the terminal arm
• Initial Arm: the starting point of the angle, it lies in the x- axis
• Terminal Arm: the arms that moves in a counterclockwise motion to make the angle
• Related acute angle: the angle between the terminal arm and the x-axis, only when the terminal arm is in quadrant 2,3, or4
• Negative Angle: an angle measured from the x- axis but in a clockwise direction
Practice Problem:
For the following diagram state the value of the principal angle and related acute angle, and state which arm (colour) represents the initial and terminal arm.
Solution:
Principal angle- 120 degrees
Related Acute angle- 60 degrees
Initial arm- red
Terminal arm- blue

## 4. Evaluating Trigonometric Ratios for Any Angle Between 0 & 360

The trigonometric ratios can be used to determine the value of the principle angle and related acute angle on a Cartesian plane. Using any point (x,y) on the terminal arm can be used to determine the angles.

The CAST rule helps you remember which trig ratios are positive within each quadrant.
Quadrant 1: ALL trig ratios are positive
MCR3U - Unit 6 - Trigonometric Ratio - 3 - CAST Rule - 2012-2013
Practice Problem:
Using the point (-8,3) determine the value of angle A.
Solution:
1. Use the point given and a primary trig ratio to find the angle of the related acute angle.
2. Subtract the related acute angle from 180 degrees to find the angle A.
tan^-1 = (3/ -8)
= - 21 degrees

A = 180 - 21
A = 159 degrees

## 5. Trigonometric Identities

Key Terms:
• Identity: equations involving trigonometric ratios that is always true for all values of the given variables.
- both the right side and left side of an equation need to equivalent to prove that it is an identity
- usually the left side needs to equal the right side in an equation, working on the left side first will help to get identical sides since the right side is usually already simplified
- rewrite the equation in terms of sine and cosine
- if there are fractions you need to use restrictions so that they don't equal zero
Trigonometric Identities: How to Derive / Remember Them - Part 1 of 3
Practice Problem:
Prove the following trigonometric identity using the trig identity chart above.
sinx = cosx
tanx

Solution:
1. Separate the left side and right side of the identity
2. Change the "tanx" on the left side in terms of sin and cos
3. Simplify the left side until it is identical to the right side

## 6. The Sine Law

Key Terms:
• Sine Law: used to measure angles or sides on triangle (either using two angles and one side, or two sides and one opposite angle)
• The ambiguous case: a case where either zero, one or two triangles are formed using the given information
• Bearing: the angle formed in a clockwise direction starting from north
The Sine Law:
Ambiguous Case:
Practice Problem:
Find the angle R using the sine law and the triangle below.
Solution:
1. Set up the sine law using the given information from the triangle
2. Isolate for the unknown, in this case angle R

sin39
/ 28 = sinR / 41
sinR = (sin 39)(41) / 28
sinR =0.92
<R = sin^-1 = 67 degrees
Law of sines

## 7. The Cosine Law

Key Terms:
• Cosine Law: formula used to determine missing sides or angles from a triangles

- you can use cosine law when you are given the measures of all three sides of the triangle

- or you could use it when you are given the measures of two sides and one angle

Practice Problem:
Using the cosine law determine the length of the missing side.
Solution:

## 8. Solving Three- Dimensional Problems by Using Trigonometry

When solving three- dimensional problems the approach you use depends on the information you are provided with in the question. There are many ways to solve the problem including the Sine Law, the Cosine Law, the Pythagorean Theorem, and the trigonometric ratios.

*
the easiest way to start a three- dimensional problem is by drawing a diagram of the situation and then labeling it with all your given information
Problem:

Using the triangle below determine the length of side h.

Solution:
1. Using triangle FED use the primary trig ratio sin46 to find the length of side ED
2. Using triangle CDE use the primary trig ratio tan38 to find the length of side h.