Learn All Trigonometric Ratios in 30 Seconds

Learn All Trigonometric Ratios in 30 Seconds

Imagine you’re standing in front of a tall tower. You can’t climb it, but you want to know its height. How do you do it? 🤔 The answer is Trigonometry!

Don’t Fear the Word

"Trigonometry = simply about ratios of sides in a right triangle."

Step 1: The Right Triangle Setup

- Every right triangle has 3 sides:

1. Hypotenuse (H) = longest side

2. Opposite (O) = side opposite to the angle θ

3. Adjacent (A) = side next to the angle θ

Step 2: The Magic Ratios 
(SOH–CAH–TOA)

sin θ = O/H = Opposite / Hypotenuse

cos θ = A/H = Adjacent / Hypotenuse

tan θ = O/A = Opposite / Adjacent

The other three are just reverse or reciprocals:

csc θ  = H/O = Hypotenuse / Opposite

sec θ = H/A = Hypotenuse / Adjacent

cot θ = A/O = Adjacent / Opposite

In other words,

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ 

Step 3: The 30-Second Memory Trick

Just remember this chant: 

SOH – CAH – TOA

- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent

That’s all you need to fix it in memory!

Step 4: Quick Example (Tower Problem) 

Problem: You are 50 m away from a tower, and the angle of elevation = 30°.

Solution:

Distance from tower = 50 m
Angle of elevation = 30°
Height of tower = ?

Formula:
tan θ = O / A = Opposite / Adjacent

So,

tan 30° = Height / 50

1/√3 = Height / 50

Height = 50 / √3 ≈ 28.9 m ✅

✅ 30-Second Recap

sin = O/H
cos = A/H
tan = O/A
csc = H/O
sec = H/A
cot = A/O

Just say “SOH–CAH–TOA” and you’ll never forget!

That is,

sin = O/H, cos = A/H, tan = O/A.

Other three are reciprocals.

SOH–CAH–TOA is the key!

About the Author

Lata Agarwal

Mathematics, Science and Astronomy professional, M.Sc. and M.Phil. in Maths with 10+ years of experience as Assistant Professor and Subject Matter Expert.

Author at Prinsli.com