### Height and Distance Formulas

### 1. Trigonometric Basics

sinθ=opposite sidehypotenuse=yrsinθ=opposite sidehypotenuse=yr

cosθ=adjacent sidehypotenuse=xrcosθ=adjacent sidehypotenuse=xr

tanθ=opposite sideadjacent side=yxtanθ=opposite sideadjacent side=yx

cscθ=hypotenuseopposite side=rycscθ=hypotenuseopposite side=ry

secθ=hypotenuseadjacent side=rxsecθ=hypotenuseadjacent side=rx

cotθ=adjacent sideopposite side=xycotθ=adjacent sideopposite side=xy

From Pythagorean theorem, x2+y2=r2x2+y2=r2 for the right angled triangle mentioned above.

### 2. Basic Trigonometric Values

θ in degrees | θ in radians | sin θ | cos θ | tan θ |

0° | 0 | 0 | 1 | 0 |

30° | π6π6 | 1212 | √3232 | 1√313 |

45° | π4π4 | 1√212 | 1√212 | 1 |

60° | π3π3 | √3232 | 1212 | √33 |

90° | π2π2 | 1 | 0 | Not defined |

### 3. Trigonometric Formulas

**Degrees to Radians and vice versa**

360°=2π radian360°=2π radian**Trigonometry Quotient Formulas**

tanθ=sinθcosθtanθ=sinθcosθ

cotθ=cosθsinθcotθ=cosθsinθ

**Trigonometry – Reciprocal Formulas**

cscθ=1sinθcscθ=1sinθ

secθ=1cosθsecθ=1cosθ

cotθ=1tanθcotθ=1tanθ

**Trigonometry – Pythagorean Formulas**

sin2θ+cos2θ=1sin2θ+cos2θ=1

sec2θ−tan2θ=1sec2θ−tan2θ=1

csc2θ−cot2θ=1csc2θ−cot2θ=1

### 4. Angle of Elevation

Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, angle of elevation is the angle between the horizontal and the line from the object to the observer’s eye (the line of sight).

i.e., angle of elevation = AOP

### 5. Angle of Depression

Suppose a man from a point O looks down at an object P, placed below the level of his eye. Then, angle of depression is the angle between the horizontal and the observer’s line of sight

i.e., angle of depression = AOP

### 6. Angle Bisector Theorem

Consider a triangle ABC as shown above. Let the angle bisector of angle A intersect side BC at a point D. Then

BDDC=ABACBDDC=ABAC

(Note that an angle bisector divides the angle into two angles with equal measures.

i.e., BAD = CAD in the above diagram)

### Height And Distance: Application In Trigonometry

The topic heights and distance is one of the applications of Trigonometry, which is extensively used in real-life. In the height and distances application of trigonometry, the following concepts are included:

- Measuring the heights of towers or big mountains
- Determining the distance of the shore from the sea
- Finding the distance between two celestial bodies

It should be noted that finding the height of bodies and distances between two objects is one of the most important applications of trigonometry.

### What are Heights and Distances?

The most significant definitions that are used when dealing with heights and distances are given as:

**Definition 1:**The line which is drawn from the eyes of the observer to the point being viewed on the object is known as the line of sight.**Definition 2:**The angle of elevation of the point on the object (above horizontal level) viewed by the observer is the angle which is formed by the line of sight with the horizontal level.**Definition 3:**The angle of depression of the point on the object (below horizontal level) viewed by the observer is the angle which is formed by the line of sight with the horizontal level.

## How to Find Heights and Distances?

To measure heights and distances of different objects, we use trigonometric ratios. For example, in fig.1, a guy is looking at the top of the lamppost. AB is horizontal level. This level is the line parallel to ground passing through the observer’s eyes. AC is known as the line of sight. ∠A is called the angle of elevation. Similarly, in fig. 2, PQ is the line of sight, PR is the horizontal level and ∠P is called the angle of depression.

Angle of Elevation in Height And Distance

Angle of Depression in Height And Distance

The angles of elevation and depression are usually measured by a device called Inclinometer or Clinometer.

Inclinometer for Measuring Angles of Elevation and Depression

**Table 1: Trigonometric Ratios of Standard Angles**

∠C | 0° | 30° | 45° | 60° | 90° |

sin C | 0 | ½ | 1/√2 | √3/2 | 1 |

cos C | 1 | √3/2 | 1/√2 | ½ | 0 |

tan C | 0 | 1/√3 | 1 | √3 | Not Defined |

cosec C | Not Defined | 2 | √2 | 2/√3 | 1 |

sec C | 1 | 2/√3 | √2 | 2 | Not Defined |

cot C | Not Defined | √3 | 1 | 1/√3 | 0 |

Solve a question based on the above topic for better understanding of this topic.

#### For More Information On Trigonometry – Measuring Heights And Distances, Watch The Below Video:

20,758

### Heights and Distances Example Question

**Question: **An aeroplane is flying h meters above the ground. At a particular instant, the angle of elevation of the plane from the eyes of a boy sitting on the ground is 60°. After some time, the angle of elevation changed to 30°. Find the distance covered by the plane during that time assuming it travelled in a straight line.

**Solution:**

The scenario explained in the question can be drawn as shown in the figure.

Height And Distance Example Question

In ∆ OAB,

tan 60° = AB/OA

√3 = h/x

x = h/√3

In ∆ OCD,

tan 30° = CD/OD

1/√3 = h/(x+y)

x + y = √3h

Distance travelled by plane = AD = y

(x + y) − x = √3h − h/√3

y = (2/√3)h

So, if the aeroplane is flying h meters above the ground, it would travel for (2/√3) h meters as the angle of elevation changes from 60° to 30 °.