Trigonometry
What is trigonometry?
Trigonometry is the branch of mathematics that studies the relationship between the sides and angles of triangles. It study about all types of triangle, but it start with a right angled triangle
Trigonometry is the branch of mathematics that studies the relationship between the sides and angles of triangles. It study about all types of triangle, but it start with a right angled triangle
What is a right angled triangle?
A right-angled triangle is a triangle that has one angle equal to \(90^o\), (a square corner), the other two angles are less than 90 degrees.
The side opposite the right angle is the longest side. We call it the hypotenuse. The other two sides are just called the legs (sometimes "base" and "height") or (sometimes "base" and "perpendicular").
A right-angled triangle is a triangle that has one angle equal to \(90^o\), (a square corner), the other two angles are less than 90 degrees.
The side opposite the right angle is the longest side. We call it the hypotenuse. The other two sides are just called the legs (sometimes "base" and "height") or (sometimes "base" and "perpendicular").
Types of Triangle
There are three types of triangle based on angle. They are
- Acute angled Triangle
All angles are less than \(90^\circ\), see figure \(\triangle ABC\)
An acute triangle is a triangle with all three angles measuring less than 90°. In other words, all the angles of an acute triangle are acute angles. - Right angled triangle
Any one angle is equal to \(90^\circ\), see figure \(\triangle PQR\)
A right triangle is a triangle in which one angle measures 90° and the other two angles measure less than 90°. In a right angled triangle, the side opposite to the right angle is called the hypotenuse while the other two sides are called legs. The hypotenuse is always the longest side while the other two sides are shorter. The below figure shows a right angled triangle PQR in which angle C measures 90°. - Obtuse angled triangle
Any one angle is greater than \(90^\circ\), see figure \(\triangle XYZ\)
An obtuse triangle is a triangle with one angle measuring more than 90° while the other two angles measure less than 90° each. In other words, an obtuse angled triangle has one obtuse angle and two acute angles. The below figure shows an obtuse angled triangle XYZ.
Test your understanding: Q1
Identify the type of triangle (based on angles):
FAQ: Reference Angle
What is reference angle?
In a right angled triangle, a reference angle is a given acute angle. It is useful to analyze trigonometric ratios. In the given triangle \(\triangle ABC\) below, \(\measuredangle A\) is reference angle.
In a right angled triangle, a reference angle is a given acute angle. It is useful to analyze trigonometric ratios. In the given triangle \(\triangle ABC\) below, \(\measuredangle A\) is reference angle.
FAQ: p,b,h
How do we identify hypoteneous , perpendicular and base?
Identifying the hypotenuse, perpendicular, and base in a right-angled triangle always depends on the location of the right angle and the reference angle, which is as below.
Identifying the hypotenuse, perpendicular, and base in a right-angled triangle always depends on the location of the right angle and the reference angle, which is as below.
- hypotenuse (h)
Always the longest side of a right-angled triangle. It is always the side opposite the right angle (90°). - perpendicular (p), alos called opposite
The side opposite the reference angle is called perpendicular - base (b), also called adjacent
The side excluding the perpendicular and hypotenuse is called base.
FAQ: Right angled triangle
How do we prove a right angled triangle?
The Pythagoras theorem states that in a right angled triangle, the square of the length of the hypotenuse (h) is equal to the sum of the squares of the other two sides (p and b). This theorem is represented by the equation
\(p^2 + b^2 = h^2 \)
So, we prove right angled triangle by showing the relation \(p^2 + b^2 = h^2 \).
The Pythagoras theorem states that in a right angled triangle, the square of the length of the hypotenuse (h) is equal to the sum of the squares of the other two sides (p and b). This theorem is represented by the equation
\(p^2 + b^2 = h^2 \)
So, we prove right angled triangle by showing the relation \(p^2 + b^2 = h^2 \).
What is pythagoras theorem?
Pythagoras theorem is
\(h^2=p^2+b^2\)
It is the relation of three sides (p,b,h) of a right angled triangle.
Test your Understanding: Q2
Q1: For ∠B: Identify perpendicular (p), base (b), and hypotenuse (h):
Q2: For ∠C:Identify perpendicular (p), base (b), and hypotenuse (h):
Trigonometric ratio
What is trigonometric ratio?
Trigonometric ratio is the 6 different ratios obtained from 3 sides of a right angled triangle.
They are given as below.
Trigonometric ratio is the 6 different ratios obtained from 3 sides of a right angled triangle.
They are given as below.
| Ratio | Name | Full Name | Ratio | Name | Full Name |
|---|---|---|---|---|---|
| \(\dfrac{p}{h}\) | \(\sin \theta\) | sine theta | \(\dfrac{h}{p}\) | \(\csc \theta\) | cosecant theta |
| \(\dfrac{b}{h}\) | \(\cos \theta\) | cosine theta | \(\dfrac{h}{b}\) | \(\sec \theta\) | secant theta |
| \(\dfrac{p}{b}\) | \(\tan \theta\) | tangent theta | \(\dfrac{b}{p}\) | \(\cot \theta\) | cotangent theta |
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Flip Card
What is the formula for pythagoras theorem?
\(h^2=p^2+b^2\)
Drag and Drop Quiz: Try it
Drag the items from the left and drop them into their correct categories on the right:
Pythagoras
Sine
Tangent
Hypotenuse
\(\frac{Opposite}{Hypotenuse}\)
Longest side
\(\frac{Opposite}{Adjacent}\)
\(p^2+b^2=h^2\)
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Multiple Choice Quiz: Try it
Question 1 of 5
Trigonometry is study about
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