Etymologically the word trigonometry is made up of two words Trigonon + metron
Trigonon means triangle while metron refers to measurement therefore,Trigonometry is the measurement of triangle.
Trigonometry is certainly crucial when it comes to finding angles and sides in a triangular shape.In real world scenarios, From Earth to space trigonometry has a wide range of applications.
Let's start the Trigonometry with an example so that you have More clarity over how it solves real world problems.
Trigonon means triangle while metron refers to measurement therefore,Trigonometry is the measurement of triangle.
Trigonometry is certainly crucial when it comes to finding angles and sides in a triangular shape.In real world scenarios, From Earth to space trigonometry has a wide range of applications.
Let's start the Trigonometry with an example so that you have More clarity over how it solves real world problems.
Example
CASE 1 : Suppose you are standing 50 m away from the base of a 500 m tall tower.you and the base of tower are on the same Plane. Now you want to find the length of the rope that connects the top of the tower with the point point you are standing at.
(The angle can be measured that the rope makes with the ground)
CASE 1 : Suppose you are standing 50 m away from the base of a 500 m tall tower.you and the base of tower are on the same Plane. Now you want to find the length of the rope that connects the top of the tower with the point point you are standing at.
(The angle can be measured that the rope makes with the ground)
CASE 2: Going with the above example ahead Suppose the length of the rope that connects top of the tower with the point you are standing and your distance from the base of tower is known then you can easily find the length of the tower.
Both the cases discussed above are applications of trigonometry. You will be able to solve them once you finish with the fundamentals.
Basics of Trigonometry
There are six trigonometric ratios namely SinA, cosA , tanA , Cosec ,secA and cotA
We will be taking a right angle triangle to define all the ratios.
\(Sin A = \frac{\text{Side opposite to angle } A}{\text{Hypotenuse}} \)
SinA = \( \frac{BC}{AC} \)
\( Cos A = \frac{\text{Side adjacent to angle } A}{\text{Hypotenuse}} \)
CosA = \( \frac{AB}{AC} \)
\( Tan A = \frac{\text{Side opposite to angle } A}{\text{Side adjacent to angle } A} \)
Tan A = \( \frac{BC}{AB} \)
The other trigonometric ratios Cosec,Sec and Cot are opposite to the above defined ratios.
\( Cosec A = \frac{1}{\sin A} = \frac{AC}{BC} \)
\( Sec A = \frac{1}{\cos A} = \frac{AC}{AB} \)
\( Cot A = \frac{1}{\tan A} = \frac{AB}{BC} \)
Conversion of Trigonometry Ratios
Sin(90-A) = CosA , Cos(90-A) = SinA
Sec(90-A) = CosecA , Cosec(90-A) = SecA
Tan(90-A) = CotA , Cot(90-A) = TanA
Cosine Rule For Non-right Angle Triangles
Cosine Rule comes in handy when a triangle is Non-right Angle or oblique.
It is used When two sides and angle between them is given or All the three sides of the triangle are given and we have to find any of the three angles of the triangle. For the triangle ABC shown at the begining of this article
\( Cos A = \frac{B^2 + C^2 - A^2}{2BC} \)
\( Cos B = \frac{A^2 + C^2 - B^2}{2AC} \)
\( Cos C = \frac{A^2 + B^2 - C^2}{2AB} \)
Trigonometric Identities :
Here are the three trigonometry identities.
\(sin^2 \theta + \cos^2 \theta = 1\)
\( 1 + \tan^2 \theta = \sec^2 \theta \)
\( 1 + \cot^2 \theta = cosec^2 \theta \)
Trigonometric Values table (0°-90° ) :
| Angle (θ) | sin θ | cos θ | tan θ | cosec θ | sec θ | cot θ |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
| 30° | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | 1 | 0 | Not defined | 1 | Not defined | 0 |
Solution For Case 1 and Case 2
Case 1 Solution :
Height of tower = 500 m
Horizontal height from the Base of tower = 50 m
Let the angle of rope from ground = 60°
In triangle ABC
Cos C = BC/AC
Cos 60° = 50/AC
AC = 50/Cos 60°. [Cos 60° = 1/2 ]
AC = 50/(1/2)
AC = 100 m
Case 2 Solution :
Suppose If the height of the tower is not given. We have to now find it.
Horizontal height from the Base of tower = 50 m
Let the angle of rope from ground = 60°
In triangle ABC
Tan C = AB/BC
Tan 60° = AB/50. [Tan 60° = √3 ]
AB = 50√3
Height of the Tower = 50√3 m
Frequently Asked questions Based on Trigonometry
Q 1 : Which trigonometry ratios have values greater than 1 ?
Ans : The value for Cosec A ,Sec A and Tan A can be greater than 1 at different angles.
Q 2 : Which trigonometry ratios have values less than or equal to 1 ?
Ans : The value for Cos A and Sin A is always less than or equal to 1 at different angles.
Q 3 : Are trigonometric identities equations?
Ans : No. Identities are always true for all values of θ.