Trigonometry is the part of mathematics in which we learn to find the base, height & angles in a triangle. It is used in different fields where we use triangular shapes.
In this article, we are going to understand what is Hypotenuse, Perpendicular, Base & Ratios in a right-angled triangle? The following points we will cover on it:
1. What is perpendicular base and hypotenuse in trigonometry?
2. How to identify base perpendicular and hypotenuse in trigonometry?
3. How to find the base perpendicular and hypotenuse?
4. What is the ratio in trigonometry?
5. How to find the ratios in trigonometry?
First we understand what is perpendicular base and hypotenuse in trigonometry?
1. What is perpendicular base and hypotenuse in trigonometry?
In any triangle, there are three sides and three vertices. Hypotenuse, base and perpendicular are the sides of a right-angled triangle. But how to define them?
a) What is perpendicular in trigonometry?
In a right triangle, there are three sides in which two sides make a 90º angle to each other. One of them is known as the Perpendicular or height of a right triangle.
b) What is a base in a triangle?
The base of a triangle is the bottom side of any triangle and sometimes it makes a 90º angle with the perpendicular or height of the triangle.
c) What is a hypotenuse in trigonometry?
The hypotenuse is the longest side of any right-angled triangle and it is opposite to the 90º angle.
Ok, now we understood how to identify them in a right-angled triangle.
2. How to identify base perpendicular and hypotenuse in trigonometry?
In below figures, the red colour shows the Perpendicular & Blue colour shows the Base and the Black colour shows the Hypotenuse.
But the question arising how to place perpendicular, base and hypotenuse in the right triangle. Let's discuss-
a) How to identify a perpendicular in the triangle?
In the figure, if we talk about angle θ so its opposite side PQ & BC are called perpendicular as shown by the arrow.
Perpendicular depends on the angle θ that we are taking.
b) How to identify the base in the triangle?
The base makes a 90º angle with perpendicular as shown by the blue colour in the above picture.
c) How to identify a hypotenuse in the triangle?
The remaining side other than the base & perpendicular will be the hypotenuse in the right triangle. It is opposite to the 90º angle.
The hypotenuse is the longest side of a right triangle.
3. How to find the base perpendicular and hypotenuse?
To find the base, perpendicular and hypotenuse we should know Pythagoras' theorem.
PYTHAGORAS THEOREM:
In a right-angled triangle square of its Hypotenuse is equal to the sum of the square of its Base and Perpendicular.
H² = P² + B²
a) How to find a Base in trigonometry?
To find the Base of the right triangle we need the values of the hypotenuse and perpendicular of that triangle. Here is the formula to find the base of triangle-
Base² = Hypotenuse² - Perpendicular²
Let's take an example for a better understanding. In right triangle hypotenuse = 5cm, Perpendicular = 4cm
Base² = Hypotenuse² - Perpendicular²
Putting all the given values-
(Base)² = (5)² - (4)²
(Base)² = 25 - 16
(Base)² = 9
(Base)² = 3²
Base = 3cm
b) How to find perpendicular in trigonometry?
To find the Perpendicular of the right triangle we need the values of the hypotenuse and Base of that triangle.
Perpendicular² = Hypotenuse² - Base²
Let we have Hypotenuse = 13cm, Base = 5cm
Perpendicular² = Hypotenuse² - Base²
Putting all the given values-
(Perpendicular)² = (13)² - (5)²
(Perpendicular)² = 169 - 25
(Perpendicular)² = 144
(Perpendicular)² = 12²
Perpendicular = 12cm
c) How to find hypotenuse in trigonometry?
To find the Hypotenuse of the right triangle we need the values of the Perpendicular and Base of that triangle.
Hypotenuse² = Perpendicular² + Base²
Let's take an example for a better understanding. We have Perpendicular = 8cm, Base = 6cm
Hypotenuse² = Perpendicular² + Base²
Putting all the given values-
(Hypotenuse)² = (8)² + (6)²
(Hypotenuse)² = 64 + 36
(Hypotenuse)² = 100
(Hypotenuse)² = 10²
Hypotenuse = 10cm
4. What is the ratio in trigonometry?
Ratios are the fractions of sides of the right triangle. According to the below figure, we are discussing the ratios with their names:
Sinθ = Perpendicular/Hypotenuse
According to the picture-
Sinθ = (PQ / PR) or (BC / AC)
Cosecθ = Hypotenuse/Perpendicular
The ratio of cosec is the inverse of sin.
According to the picture-
Cosecθ = (PR / PQ) or (AC / BC)
Cosθ = Base/Hypotenuse
According to the picture-
Cosθ = (QR / PR) or (AB / AC)
Secθ = Hypotenuse/Base
The ratio of sec is the inverse of cos.
According to the picture-
Secθ = (PR / QR) or (AC / AB)
Tanθ = Perpendicular/Base
According to the picture-
Tanθ = (PQ / QR) or (BC / AB)
Cotθ = Base/Perpendicular
The ratio of the cot is the inverse of tan.
According to the picture-
Cotθ = (QR / PQ) or (AB / BC)
5. How to find the ratios in trigonometry?
Example:-
In a triangle, right angled at B. BC = 3 cm & AB = 4 cm. Find all the trigonometric ratios sin, cos, tan, cosec, sec, and cot.
Solve:-
Given, BC = 3cm, AB = 4cm, AC =?
To find other trigonometric ratios we need hypotenuse. So we are using Pythagoras' Theorem -
H² = P² + B²
From the above figure for angle θ we have-
(AC)² = (BC)² + (AB)²
Putting all the given values-
(AC)² = (3)² + (4)²
(AC)² = 9 + 16
(AC)² = 25
(AC)² = 5²
AC = 5 cm.
Now we have all the values for finding the ratios -
Perpendicular(BC) = 3cm,
Base(AB) = 4cm &
Hypotenuse(AC) = 5 cm.
Sinθ = P/H
= BC / AC
= 3 / 5
Cosecθ = H/P
= AC / BC
= 5 / 3
Cosθ = B/H
= AB / AC
= 4 / 5
Secθ = H/B
= AC / BC
= 5 / 4
Tanθ = P/B
= BC / AB
= 3 / 4
Cotθ = B/P
= AB / BC
= 4 / 3
So finally we got all the trigonometric ratios.
