# Relations Between the Trigonometric Ratios

Fundamental relations between the 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)

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.