Introduction to Trigonometry
Mathematics Class Tenth
Important formula trigonometric ratios
Trigonometry is the study of relationships between the sides and angles of a triangle.
The word 'Trigonometry' is the combination of three Greek words 'Tri + Gon + Metron'.
In which 'Tri' means three, and
'Gon' means sides and
'Metron' means to measure.
Thus, the full meaning of word 'Trigonometry' is 'Measuring the three sides' of a triangle.
Using the principles and methods of Trigonometry, the height and distance of many big objects can be calculated after the imagination of the right triangle is formed by observation of those images.
Example: While visiting Qutub Minar if a student is looking at the top of the Minar, a right angle can be imagined to be made.
It is possible to know the height of the Qutub Minar without actually measuring it by applying principles and techniques of Trigonometry.
The earliest known work on Trigonometry was recorded in Egypt and Babylon. Early astronomers used Trigonometry to find out the distances of the stars and planets from the Earth.
Right Angle Triangle
A triangle with an angle equal to 90o is called Right Angled Triangle or simply Right Triangle.
In a Right Angled Triangle, there are two acute angles (angles less than 90o) and one right angle.
Hypotenuse
(a) The side opposite to right angle (90o) is called Hypotenuse in a right-angled triangle.
(b) Hypotenuse is the largest side of a right-angle triangle.
(c) But the length of the hypotenuse is less than the sum of the rest of the two other sides.
(d) Hypotenuse is generally denoted by the letter "h".
Base
(a) The side adjacent to one of the acute angles, (say ∠ A in the given figure) is called Base. Generally, the side which forms the base of the right-angled triangle other than the hypotenuse is called the base.
(b) Base is generally denoted by the letter "b".
Perpendicular
(a) The side opposite to the acute angle, (say ∠ A in the given figure) is called Perpendicular.
(b) The height is generally denoted by letter "p".
Pythagoras Theorem
Pythagoras Theorem gives the relationship among the sides of a right-angled triangle.
According to Pythagoras Theorem, In a right-angled triangle
(Hypotenuse) 2 = (perpendicular) 2 + (base) 2
⇒ h2 = p2 + b2
Where, h = hypotenuse, p = perpendicular and b = base.
Trigonometric Ratio
Relation among acute angle and sides of a right angled triangle is called Trigonometric Ratio.
Let, the acute angle is "A".
Thus, sine of ∠A = BC/AB = p/h
In short, sine of ∠ A is written as "sin A".
Here "sin A" is not "sin" multiplied by "A". Rather it is the sine of ∠ A
sin A = p/h
cos A = b/h
tan A = p/b
cosec A = h/p
sec A = h/b
cot A = b/p
Where,
p = perpendicular
b = base
h = hypotenuse
Trick to remember trigonometric Ratios
"Pandit Badri Prasad, Har Har Bol ".
Get this line by heart.
And draw out the first letter of every word given in the line. For convenience first letter of every word is highlighted in red.
This becomes, "PBP : HHB"
Now, we have two words.
And we have to find out, (1) sin, (2) cos, and (3) tan of the acute angle.
(1) Now, for "sin A" make the ratio of first letters from both the words,
"PBP : HHB"
i.e. P and H.
This becomes, sin A = p/h
(2) For "cos A"
Make ratios of second letters from each of the words,
"PBP : HHB"
This becomes, B and H
Thus, cos A=b/h
(3) For "tan A"
Make ratios of third means last letters from each of the words,
"PBP : HHB"
This becomes, P and B
Thus, tan A = p/b
Thus, sequence is "sin, cos, tan" and "PBP : HHB".
Ratios for "cosec, sec and cot"
"cosec" is inverse of "sin".
Thus, here we have sin A = p/h
Hence, cosec A = h/p (just opposite of "sin")
Similarly,
"sec" is the reverse of "cos"
And we have cos A = b/h
Thus, sec A = h/b [just opposite of "cos"]
Similarly,
"cot" is the reverse of "tan"
And we have, tan A = p/b
Thus, cot A = b/p [just opposite to "tan"]
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