Reciprocal Relations of Trigonometric Ratios

Reciprocal relations of trigonometric ratios are explained here to represent the relationship between the three pairs of trigonometric ratios as well as their reciprocals.

Let OMP be a right angled triangle at M and ∠MOP = θ.

Reciprocal Relations of Trigonometric Ratios

According to the definition of trigonometric ratios we have,

● sin θ = perpendicular/hypotenuse = MP/PO ………….. (i)

and csc θ = hypotenuse/perpendicular = PO/MP ………….. (ii)

From (i) sin θ = 1/(PO/MP)

⇒ sin θ = 1/csc θ ………………… (A)

Again, from (ii) csc θ = 1/(MP/PO)

⇒ csc θ = 1/sin θ ………………… (B)

From (A) and (B) we conclude that

sin θ and csc θ are reciprocal of each other.

● cos θ = adjacent/hypotenuse = OM/OP ………….. (iii)

and sec θ = hypotenuse/adjacent = OP/OM ………….. (iv)

From (iii) cos θ = 1/(OP/OM)

⇒ cos θ = 1/sec θ ………………… (C)

Again, from (iv) sec θ = 1/(OM/OP)

⇒ sec θ = 1/cos θ ………………… (D)

From (C) and (D) we conclude that

cos θ and sec θ are reciprocal of each other.

● tan θ = perpendicular/adjacent = MP/OM ………….. (v)

and cot θ = adjacent/perpendicular = OM/MP ………….. (vi)

From (v) tan θ = 1/(OM/MP)

⇒ tan θ = 1/cot θ ………………… (E)

Again, from (vi) cot θ = 1/(MP/OM)

⇒ cot θ = 1/tan θ ………………… (F)

From (E) and (F) we conclude that

 tan θ and cot θ are reciprocal of each other.

To find values of trig functions we can use these reciprocal relationships to solve different types of problems.

Note:

From the above discussion about the reciprocal trigonometric functions  we get;

1. sin θ  ∙ csc θ = 1

2. cos θ ∙ sec θ = 1

3. tan θ ∙ cot θ  = 1

● Trigonometric Functions

• Basic Trigonometric Ratios and Their Names
• Restrictions of Trigonometrical Ratios
• Reciprocal Relations of Trigonometric Ratios
• Quotient Relations of Trigonometric Ratios
  
• Limit 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
• Trigonometrical Ratios of 0°
• Trigonometrical Ratios of 30°
• Trigonometrical Ratios of 45°
• Trigonometrical Ratios of 60°
• Trigonometrical Ratios of 90°
• Trigonometrical Ratios Table
• Problems on Trigonometric Ratio of Standard Angle
  
• Trigonometrical Ratios of Complementary Angles
• Rules of Trigonometric Signs
• Signs of Trigonometrical Ratios
  
• All Sin Tan Cos Rule
• Trigonometrical Ratios of (- θ)
• Trigonometrical Ratios of (90° + θ)
• Trigonometrical Ratios of (90° - θ)
• Trigonometrical Ratios of (180° + θ)
• Trigonometrical Ratios of (180° - θ)
• Trigonometrical Ratios of (270° + θ)
• Trigonometrical Ratios of (270° - θ)
• Trigonometrical Ratios of (360° + θ)
• Trigonometrical Ratios of (360° - θ)
• Trigonometrical Ratios of any Angle
• Trigonometrical Ratios of some Particular Angles
• Trigonometric Ratios of an Angle
• Trigonometric Functions of any Angles
• Problems on Trigonometric Ratios of an Angle
• Problems on Signs of Trigonometrical Ratios

10th Grade Math

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