Relations Between the
Trigonometric Ratios

Fundamental relations between the trigonometric ratios of an angle:

Trigonometric Ratios of an Angle

To know the relations between the trigonometric ratios from the above figure, we see;

sin θ = perpendicular/hypotenuse = MP/PO and

cosec θ = hypotenuse/perpendicular = PO/MP

It is clear that one is the reciprocal of the other.

So, sin θ = 1/cosec θ and

cosec θ = 1/sin θ ………. (a)

Again, cos θ = base/hypotenuse = OM/OP and

sec θ = hypotenuse/ base = OP/OM

One is reciprocal of the other.

That is, cos θ = 1/sec θ and sec θ = 1/cos θ ………. (b)

So, tan θ = perpendicular/base = MP/OM and cot θ = base/perpendicular = OM/MP

tan θ = 1/cot θ and cot θ = 1/tan θ ………. (c)

Moreover, sin θ/cos θ = (MP/OP) ÷ (OM/OP) = (MP/OP) × (OP/OM) = MP/OM = tan θ

Therefore, sin θ/cos θ = tan θ ………. (d)

and cos θ/sin θ = (OM/OP) ÷ (MP/OP) = (OM/OP) × (OP/MP) = OM/MP = cot θ

Therefore, cos θ/sin θ = cot θ ………. (e)

relations between the trigonometric ratios
Sin θ = PM/OP

Cos θ = OM/OP

Tan θ = PM/OM

Csc θ = OP/PM

Sec θ = OP/OM

Cot θ = OM/PM



Now from the right-angled triangle POM we get;

PM2 + OM2 = OP2 ……………. (i)

Dividing both sides by OP2 we get,

PM2/OP2 + OM2/OP2 = OP2/OP2

or, (PM/OP)2 + (OM/OP)2 = 1

or, sin2 θ + cos2 θ = 1

Again, dividing both sides of (i) by OM2

PM2/OM2 + OM2/OM2 = OP2/OM2

or, (PM/OM)2 + 1 = (OP/OM)2

or, tan2 θ + 1 = sec2 θ

Finally, dividing both of (i) by PM2 we get;

PM2/PM2 + OM2/PM2 = OP2/PM2

or, 1 + (OM/PM)2 = (OP/PM)2

or, 1 + cot2 θ = csc2 θ


Corollary 1: From the relation sin2 θ + cos2 θ = 1 we deduce that

(i) 1 - cos2 θ = sin2 θ and

(ii) 1 - sin2 θ = cos2 θ


Corollary 2: From the relation 1 + tan2 θ = sec2 θ we deduce that

(i) sec2 θ - 1 = tan2 θ and

(ii) sec2 θ - tan2 θ = 1


Corollary 3: From the relation 1 + cot2 θ = csc2 θ we deduce that

(i) csc2 θ - 1 = cot2 θ and

(ii) csc2 θ - cot2 θ = 1


This is how the ratios are related to show that one is the reciprocal of the other according to the relations between the trigonometric ratios.

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 Relations Between the 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. Worksheet on Area of a Square and Rectangle | Area of Squares & Rectan

    Jul 19, 25 05:00 AM

    Area and Perimeter of Square and Rectangle
    We will practice the questions given in the worksheet on area of a square and rectangle. We know the amount of surface that a plane figure covers is called its area. 1. Find the area of the square len…

    Read More

  2. Area of Rectangle Square and Triangle | Formulas| Area of Plane Shapes

    Jul 18, 25 10:38 AM

    Area of a Square of Side 1 cm
    Area of a closed plane figure is the amount of surface enclosed within its boundary. Look at the given figures. The shaded region of each figure denotes its area. The standard unit, generally used for…

    Read More

  3. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  4. Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers

    Jul 17, 25 12:40 AM

    Most and Least Perimeter
    Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find…

    Read More

  5. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 11:46 PM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More