What are Reciprocal Ratios?
Reciprocal trigonometric ratios are obtained by taking the reciprocal of the three basic ratios:sin θ,cos θand tan θ.
This gives us three more important ratios:cosec θ,sec θand cot θ.
A reciprocal ratio is simply the reciprocal of another trigonometric ratio.
The three important ratios
There are three reciprocal trigonometric ratios:cosec θ,sec θand cot θ.
Remember the pairs
Each basic trigonometric ratio has one reciprocal ratio. Memorise these three pairs carefully.
Using the sides of a triangle
Just like sin, cos and tan, reciprocal ratios can also be expressed using the sides of a right-angled triangle.
Three formulas to remember
cosec θ
sec θ
cot θ
Think of them as pairs
Test Your Understanding
Finding Secant and Product of Tan & Cot
If cos θ = 5/13, find the value of sec θ and evaluate (tan θ × cot θ).
Solution:
- sec θ = 1 / cos θ = 1 / (5/13) = 13/5
- Since cot θ is the reciprocal of tan θ, tan θ × cot θ = tan θ × (1 / tan θ) = 1
Evaluating Cosec A + Sec A
Given tan A = 4/3, evaluate (cosec A + sec A).
Solution:
tan A = Perpendicular / Base = 4/3 → Hypotenuse = √(4² + 3²) = 5
- sin A = 4/5 → cosec A = 5/4
- cos A = 3/5 → sec A = 5/3
= (15 + 20) / 12 = 35/12
What did we learn?
cosec θ is the reciprocal of sin θ.
sec θ is the reciprocal of cos θ.
cot θ is the reciprocal of tan θ.
The three reciprocal ratios arecosec θ, sec θ and cot θ.