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