# Trigonometry

### By: Shera.P

## Topics

- Primary and Reciprocal Trigonometric Ratios
- Evaluating Trigonometric Ratios for Special Angles
- Exploring Trigonometric Ratios fro Angles Greater than 90
- Evaluating Trigonometric Ratios for Any Angle Between 0 and 360
- Trigonometric Identities
- The Sine Law
- The Cosine Law
- 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__ :

**Determine the corresponding reciprocal trigonometric ratio.**

__Practice Problem:__

- cosA = 4/13
- tanA = 12/25
- sinA = 2/ 17

**Solution:**

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

- secA = 13/4
- cotA = 25/12
- 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.

__Evaluate the following without using a calculator. Do not use decimals.__

**Practice Problem:**2 sin 30˚ + 3 cos 60˚ – 3 tan 45˚

__Solution:__- Use the chart to determine the values of each angle.
- Start subbing in the values into the equation.
- 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

## 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

__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.__

**Practice Problem:**__Principal angle- 120 degrees__

**Solution:**Related Acute angle- 60 degrees

Initial arm- red

Terminal arm- blue

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

The

**CAST**rule helps you remember which trig ratios are positive within each quadrant.

Quadrant 1: ALL trig ratios are positive

Quadrant 2: Sine is positive

Quadrant 3: Tan is positive

Quadrant 4: Cosine is positive

__Practice Problem:__Using the point (-8,3) determine the value of angle A.

**Solution:**- Use the point given and a primary trig ratio to find the angle of the related acute angle.
- Subtract the related acute angle from 180 degrees to find the angle A.

= - 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.

- 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

__Prove the following trigonometric identity using the trig identity chart above.__

**Practice Problem:**__sinx__= cosx

tanx

**Solution:**- Separate the left side and right side of the identity
- Change the "tanx" on the left side in terms of sin and cos
- 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:**

## Case #1 Side c is smaller than h so no triangles are formed | ## Case #2 Side c is equal to h so one right triangle is formed | ## Case #3 Side c is larger than h but smaller than side b (h < c < b) so two triangles are formed |

## Case #4 Side b is larger than side c so one triangle is formed | ## Case #5 Side c is smaller than side b so no triangles are formed | ## Case #6 Side c is larger than side b so one triangle is formed |

**Practice Problem:**

Find the angle R using the sine law and the triangle below.

__1. Set up the sine law using the given information from the triangle__

**Solution:**

2. Isolate for the unknown, in this case angle R

__/ 28 =__

**sin39**

__sinR__/ 41

sinR = (sin 39)(41) / 28

sinR =0.92

<R = sin^-1 = 67 degrees

## 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

**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:**

- Using triangle FED use the primary trig ratio sin46 to find the length of side ED
- Using triangle CDE use the primary trig ratio tan38 to find the length of side h.