Basic Trigonometric Ratios

To know about the basic trigonometric ratios with respect to a right-angled triangle,

Trigonometric Ratios

let a ray OA revolve in the anti-clockwise direction and assume the position OA1, so that an angle ∠AOA1 = θ is formed.

Now any number of points P, Q, R, .......... are taken on OA1, and perpendiculars PX, QY, RZ, ........ are drawn on OA from those points respectively.

All the right-angled triangles POX, QOY, ROZ, ......... are similar to each other.

Now from the properties of similar triangles we know,

(i) PX/OP = QY/OQ = RZ/OR = .....

(iii) PX/OX = QY/ OQ = RZ/OZ = .....

(v) OP/OX = OQ/OX = OR/OZ = .....

(ii) OX/OP = QY/OQ = OZ/OR = .....

(iv) OP/PX = OQ/QY = OR/RZ = .....

(vi) OX/PX = OY/QY = OZ/RZ = .....

Thus we see in a set of similar right-angled triangles with respect to the same acute angle

(i) perpendicular : hypotenuse i.e., perpendicular/hypotenuse remains same.

(ii) base : hypotenuse and

(iii) perpendicular : base do not change for the aforesaid similar right-angled triangles. So we can say that the values of these ratios do not depend on the size of triangles or the length of their sides. The values entirely depend on the magnitude of the acute angle θ.

It is so because all the triangles are right angled triangles having a common acute angle θ. Similar relations will hold whatever be the measure of the acute angle θ.

So we see that in similar right-angled triangles the ratio of any two sides, with reference to a common acute angle, give a definite value. This is the concept on the basis trigonometric ratios.

Again we have shown that the ratio of any two sides of a right-angled triangle, have six different ratios.

These six ratios are identified by six different names, one for each.

Now we will define trigonometrical ratios of positive acute angles and their relations.

Definitions of Trigonometrical Ratios

Definitions of Trigonometrical Ratios:

Let a revolving line OY rotates about O in the anti-clockwise sense and starting from the initial position OX comes in the final position OY and traces out an angle ∠XOY = θ where ϴ is acute. Take a any point P on OY and draw PM perpendicular to OX. Clearly, POM is a right-angled triangle. With respect to the angle θ we shall call the sides, OP, PM and OM of the ∆POM as the hypotenuse, opposite side is also known as the perpendicular and adjacent side is also known as the base.

Now, the six trigonometrical ratios of the angle θ are defined as follows:


What are the six trigonometrical ratios?

Perpendicular/Hypotenuse = PM/OP = sine of the angle θ;

                       or, sin θ = PM/OP

Adjacent/Hypotenuse = OM/OP = cosine of the angle θ;

                 or, cos θ = OM/OP

Perpendicular/Adjacent = PM/OM = tangent of the angle θ;

                   or, tan θ = PM/OM

Hypotenuse/Perpendicular = OP/PM = cosecant of the angle θ;

                      or, csc θ = OP/PM

Hypotenuse/Adjacent = OP/OM= secant of the angle θ;

                 or, sec θ = OP/OM

and Adjacent/Perpendicular = OM/PM = cotangent of the angle θ;

                         or, cot θ = OM/PM

The six ratios sin θ, cos θ, tan θ, csc θ, sec θ and cot θ are called Trigonometrical Ratios of the angle θ.

Sometimes there are two other ratios in addition. They are known as Versed sine and Coversed sine.

 These two ratios are defined as follows:

 Versed sine of angle θ or Vers θ = 1 - cos θ  
and Coversed sine of angle
θ or Coverse θ = 1 - sin θ.

Note:

(i) Since each trigonometrical ratio is defined as the ratio of two lengths hence each of them is a pure number.


(ii) Note that sin
θ does not imply sin × θ; in fact, it represents the ratio of perpendicular and hypotenuse with respect to the angle θ of a right-angled triangle.


(iii) In a right-angled triangle the side opposite to right-angle is the hypotenuse, the side opposite to given angle
θ is the perpendicular and the remaining side is the adjacent side.

Basic Trigonometric Ratios 

Relations Between the Trigonometric Ratios

Problems on Trigonometric Ratios

Reciprocal Relations of Trigonometric Ratios

Trigonometrical Identity

Problems on Trigonometric Identities

Elimination of Trigonometric Ratios 

Eliminate Theta between the equations

Problems on Eliminate Theta 

Trig Ratio Problems

Proving Trigonometric Ratios

Trig Ratios Proving Problems

Verify Trigonometric Identities 






10th Grade Math

From Basic Trigonometric Ratios to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Addition of Decimals | How to Add Decimals? | Adding Decimals|Addition

    Apr 24, 25 01:45 AM

    Addition of Decimals
    We will discuss here about the addition of decimals. Decimals are added in the same way as we add ordinary numbers. We arrange the digits in columns and then add as required. Let us consider some

    Read More

  2. Addition of Like Fractions | Examples | Videos | Worksheet | Fractions

    Apr 23, 25 09:23 AM

    Adding Like Fractions
    To add two or more like fractions we simplify add their numerators. The denominator remains same. Thus, to add the fractions with the same denominator, we simply add their numerators and write the com…

    Read More

  3. Subtraction | How to Subtract 2-digit, 3-digit, 4-digit Numbers?|Steps

    Apr 23, 25 12:41 AM

    Subtraction Example
    The answer of a subtraction sum is called DIFFERENCE. How to subtract 2-digit numbers? Steps are shown to subtract 2-digit numbers.

    Read More

  4. Subtraction of 4-Digit Numbers | Subtract Numbers with Four Digit

    Apr 23, 25 12:38 AM

    Properties of Subtraction of 4-Digit Numbers
    We will learn about the subtraction of 4-digit numbers (without borrowing and with borrowing). We know when one number is subtracted from another number the result obtained is called the difference.

    Read More

  5. Subtraction with Regrouping | 4-Digit, 5-Digit and 6-Digit Subtraction

    Apr 23, 25 12:34 AM

     Subtraction of 5-Digit Numbers with Regrouping
    We will learn subtraction 4-digit, 5-digit and 6-digit numbers with regrouping. Subtraction of 4-digit numbers can be done in the same way as we do subtraction of smaller numbers. We first arrange the…

    Read More