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 6 Times Table | Printable Multiplication Table | Video

    Mar 16, 25 03:09 PM

    worksheet on multiplication of 6 times table
    Printable worksheet on 6 times table can be used from everywhere. Homeschoolers can also use these multiplication table sheets to practice at home.

    Read More

  2. Worksheet on 5 Times Table | Printable Multiplication Table | Video

    Mar 16, 25 01:25 PM

    worksheet on multiplication of 5 times table
    Printable worksheet on 5 times table can be used from everywhere. Homeschoolers can also use these multiplication table sheets to practice at home.

    Read More

  3. Worksheet on 3 Times Table | Printable Multiplication Table | Video

    Mar 16, 25 12:59 PM

    worksheet on multiplication of 3 times table
    Printable worksheet on 3 times table can be used from everywhere. Homeschoolers can also use these multiplication table sheets to practice at home.

    Read More

  4. 5th Grade Math Worksheets | 5th Grade Homework Sheets | Math Worksheet

    Mar 16, 25 12:50 PM

    5th grade math worksheets is carefully planned and thoughtfully presented on mathematics for the students. Teachers and parents can also follow the worksheets to guide the students.

    Read More

  5. Fundamental Operations on Large Numbers Worksheet | 5th Grade Numbers

    Mar 16, 25 11:37 AM

    Fundamental Operations on Large Numbers Worksheet
    fundamental operations on large numbers worksheet

    Read More