Trigonometric Ratios: Trigonometry is the branch of Mathematics that deals with the lengths of the sides of a triangle and the angles between them. The relationship between the angles and the sides of the triangles are expressed using trigonometric functions or trigonometric ratios.
The most commonly used trigonometric functions are sine, cosine, tangent, and the reciprocals of these three functions – cosecant, secant, and cotangent. These trigonometric ratios/functions are abbreviated as sin, cos, tan, cosec, sec, and cot. In order to master trigonometry, one should have a clear understanding of what these trigonometric ratios/functions mean. In this article, we will learn about them in detail.
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Trigonometric Ratios Definition: Trigonometric Ratios (Functions) & What They Mean
Trigonometric ratios are quite important not only for students but also in day-to-day lives. A lot of professions make use of these functions to do their work. There are six commonly used trigonometric functions, i.e. sin, cos, tan, cosec, sec, and cot; which are also known as trigonometric ratios.
This is because these functions are basically the ratio of the sides of the triangle. Let’s understand these ratios.
What Is Trigonometric Ratios: Understanding Trigonometric Functions/Ratios
Let us consider the following right-angled triangle:As you can see, the three sides of the triangle are:
a. Base: The side that is horizontal to the plane.
b. Perpendicular: The side making an angle of 90 degree with the Base.
c. Hypotenuse: The longest side of the triangle opposite to the right angled vertex.
Also, X is the angle made by Hypotenuse and Base.
Then:
a) sine of angle X = sinX = \(\frac{Perpendicular}{Hypotenuse}\) |
b) cosine of angle X = cos X = \(\frac{Base}{Hypotenuse}\) |
c) tangent of angle X = tan X = \(\frac{Perpendicular}{Base}\) |
Also, we know that:
d) cotangent of angle X = cot X = \(\frac{1}{tanX}\); Therefore, cot X = \(\frac{Base}{Perpendicular}\) |
e) cosecant of angle X = cosec X = \(\frac{1}{\sin X}\); Therefore, cosec X = \(\frac{Hypotenuse}{Perpendicular}\) |
f) secant of angle X (full form of sec in trigonometry is secant)= sec X = \(\frac{1}{\cos X}\); Therefore, sec X = \(\frac{Hypotenuse}{Base}\) |
So, now you know what the six trigonometric ratios exactly mean.
Note: In the above figure, if we consider the third angle (i.e. the angle between Hypotenuse and Perpendicular, then the Base will become Perpendicular and Perpendicular will become Base and the values should be put accordingly to calculate the trigonometric ratios of that angle.
How To Memorize The Trigonometric Functions?
The easiest way to remember what the trigonometric functions mean, i.e. the ratio of which sides correspond to which trigonometric function is by using mnemonic.
Some People Have Curly Black Hair Thickly Plastered Back.
Look at the bold, underlined, first letter of each of the words and what they represent:
S of Some ↠ Sine P of People ↠ Perpendicular H of Have ↠ Hypotenuse This gives sinX (S) = \(\frac{Perpendicular (P)}{Hypotenuse (H)}\) |
C of Curly ↠ Cosine B of Black ↠ Base H of Hair ↠ Hypotenuse This gives cosX (C) = \(\frac{Base (B)}{Hypotenuse (H)}\) |
T of Thickly ↠ Tangent P of Plastered ↠ Perpendicular B of Back ↠ Base This gives tanX (T) = \(\frac{Perpendicular (P)}{Base (B)}\) |
As cosecant, secant, and cotangent are just reciprocals of the sine, cosine, and tangent functions, all you have to do is take the reciprocals of the respective ratios.
Important FAQs About Trigonometric Ratios
Given below are the important FAQs related to Trigonometric functions:
Q1: What are the trigonometric ratios?
Ans: Trigonometric ratios express the relationship between the angles and the sides of the triangles.
Q2: Do trigonometric functions only work for right triangles?
Ans: Trigonometric Functions are specifically defined for the right triangles. However, this doesn’t mean it only works for right-angled triangles. There are laws of sines and cosines that works for non-right triangles.
Q3: How do we convert one trigonometric ratio into another?
Ans: For example,
⇒ sin^{2} x + cos^{2} x = 1
⇒ sin^{2} x = 1 – cos^{2} x
⇒ sin x = √(1 – cos^{2}x)
Check out this article on Trigonometry Formulas.
Q4: How to remember the trigonometric ratios?
Ans: You can remember the simple mnemonic given in this article to remember the basic trigonometric ratios like sin, cos and tan – “Some People Have Curly Black Hair Thickly Plastered Back“.
Q5. What are the 6 trigonometric ratios?
Ans: The six trigonometric ratios are sine, cosine, tangent, cosecant, secant and cotangent, abbreviated as sin, cos, tan, csc, sec, cot.
Q6. What are the 3 trigonometric ratios?
Ans: There are three basic trigonometric ratios are sine, cosine, and tangent
Q7. Who is the father of trigonometry?
Ans: Hipparchus of Nicaea is the father of Trigonometry.
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So, now you know what the different trigonometric ratios mean. You also know how to memorize them in a manner that you will never forget. We hope it helps you.
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