# 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.

Problems on Trigonometric Ratios

Reciprocal Relations of Trigonometric Ratios

Trigonometrical Identity

Problems on Trigonometric Identities

Eliminate Theta between the equations

Trig Ratio Problems

Proving Trigonometric Ratios

Trig Ratios Proving Problems