# Trigonometry in Right-Angled Triangles

by Michelle Lim
# 1. Terminology of right-angled Triangles

## 1.1. Sides

### 1.1.1. Opposite (a)

1.1.1.1. Side opposite angle

### 1.1.2. Adjacent (b)

1.1.2.1. Side beside angle (but not hypotenuse)

### 1.1.3. Hypotenuse (c)

1.1.3.1. Longest side of triangle (right-angled)

## 1.2. Angles of Elevation and Depression

### 1.2.1. Elevation

1.2.1.1. The angle of elevation is the horizontal line up to the line of sight.

### 1.2.2. Depression

1.2.2.1. The angle of depression is the acute angle measured down from the horizontal line to the line of sight.

### 1.2.3. Line of sight

1.2.3.1. An imaginary line from the eye to the perceived object.

1.2.3.2. Given the same line of sight, the angles of elevation and depression are the same.

# 2. Trigonometrical Ratios

## 2.1. Ratios

### 2.1.1. Tangent (tan)

2.1.1.1. TOA: Opposite/ Adjacent. The ratio is known as the tangent of angle a.

### 2.1.2. Cosine (cos)

2.1.2.1. CAH: Adjacent/ Hypotenuse. The ratio is known as the cosine of angle a.

### 2.1.3. Sine (sin)

2.1.3.1. SOH: Opposite/ Hypotenuse. The ratio is known as the sine of angle a.

### 2.1.4. TOA CAH SOH: 'Big Fat Lady' in Hokkien

## 2.2. Concepts

### 2.2.1. Mathematical Concepts

2.2.1.1. Constancy

2.2.1.1.1. With the angle being fixed, the equality of value of each trigonometrical ratio is maintained regardless of size of triangle.

2.2.1.2. Patterns

2.2.1.2.1. By recognizing and understanding patterns, we can make logical deductions and justify our conclusion.

2.2.1.3. Relationships

2.2.1.3.1. Trigonometrical ratios depict the relationship amongst the sides and angles of a triangle.

# 3. Trigonometry

## 3.1. Definition

### 3.1.1. A branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides.

## 3.2. Application

### 3.2.1. Ancient times: Used in measurement of heights and distances of objects that could not be otherwise measured (Eg. distance of stars from Earth)

### 3.2.2. Present: Making quick and simple calculations regarding height and distances of far away objects- (INDIRECT MEASUREMENT)

# 4. Pythagoras' Theorem

## 4.1. Theorems

### 4.1.1. 1. Pythagoras' Theorem:

4.1.1.1. In a right-angled triangle, the square of the hypothenuse is equal to the sum of squares of the other two sides.

4.1.1.1.1. Proof: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse.

### 4.1.2. 2. Converse of Pythagoras' Theorem:

4.1.2.1. In a triangle, if the square of the longest side is equal to the sum of the squares of the remaining two sides, then the angle opposite to the longest side, is a right angle.

4.1.2.1.1. Proof

## 4.2. Concepts

### 4.2.1. Mathematical Concepts

4.2.1.1. Constancy

4.2.1.1.1. The equality of the equation representative of Pythagoras' Theorem, does not change regardless of the size of the triangle.

4.2.1.2. Relationship

4.2.1.2.1. Pythagoras' Theorem is a relationship of the size of a right-angled triangle.

4.2.1.3. Shapes

4.2.1.3.1. Pythagoras' Theorem is a geometric representation of an algebraic relation.

### 4.2.2. Macroconcepts

4.2.2.1. Models

4.2.2.1.1. Pythagoras' Theorem can be represented geometrically and be used to solve problems involving 2-dimensional and 3-dimensional models, to solve real life problems

4.2.2.1.2. The converse of Pythagoras' Theorem facilitates testing if a triangle is a right-angled triangle.

# 5. Example Question

## 5.1. Find the unknown angles and sides marked x, y and z in the figure.

## 5.2. Working