Trigonometric ratios form the base of trigonometry and help describe how the sides of a right-angled triangle relate to its angles. These ratios make it easy to calculate heights, distances, slopes, and angles in geometry and real-life measurements. Whether solving a classroom problem or analysing motion in physics, these ratios simplify everything involving angles and sides.
Trigonometric Ratios Formula Overview
| Ratio | Formula | Meaning / Relationship | When It Is Used |
|---|---|---|---|
| sin θ | Opposite / Hypotenuse | Measures vertical rise | Geometry, waves, heights & distances |
| cos θ | Adjacent / Hypotenuse | Measures horizontal run | Vector and motion problems |
| tan θ | Opposite / Adjacent | Ratio of height to base | Angle of elevation/depression |
| cosec θ | 1 / sin θ | Reciprocal of sine | Advanced trig functions |
| sec θ | 1 / cos θ | Reciprocal of cosine | Physics, light, reflection |
| cot θ | 1 / tan θ | Reciprocal of tangent | Geometry and ratios |
What are Trigonometric Ratios in Maths?
In mathematics, trigonometric ratios describe the relationship between the sides of a right-angled triangle and one of its acute angles. They are the core of trigonometry, used to connect angles with lengths. The three sides, Opposite, Adjacent, and Hypotenuse, together help define all six ratios.
To apply these ratios:
- Identify the angle θ
- Determine its opposite, adjacent, and hypotenuse sides
- Use the correct ratio depending on the sides you know
If opposite and hypotenuse are known → use sin θ
If adjacent and hypotenuse are known → use cos θ
These ratios are widely used in geometry, architecture, physics, engineering, astronomy, and exams like CUET, JEE, SSC, and Board exams.
Examples to Calculate Trigonometric Ratios
Example 1: Find sin θ and cos θ when Opposite = 3 cm and Hypotenuse = 5 cm
Step 1: sin θ = Opposite / Hypotenuse = 3/5 = 0.6
Step 2: Adjacent = √(5² − 3²) = √16 = 4
Step 3: cos θ = Adjacent / Hypotenuse = 4/5 = 0.8
Answer: sin θ = 0.6, cos θ = 0.8
Example 2: If tan θ = 3/4, find sin θ and cos θ
Step 1: Opposite = 3, Adjacent = 4 → Hypotenuse = √(3² + 4²) = 5
Step 2: sin θ = Opposite / Hypotenuse = 3/5
Step 3: cos θ = Adjacent / Hypotenuse = 4/5
Answer: sin θ = 3/5, cos θ = 4/5
FAQs about Trigonometric Ratios Formula
Q1. What are the six trigonometric ratios?
sin, cos, tan, cosec, sec, and cot.
Q2. Why are they called “ratios”?
Because each represents the ratio of two sides of a right triangle.
Q3. How can I easily remember them?
Use the rule SOH-CAH-TOA:
Sine = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.
Q4. What is the range of sin θ and cos θ?
Both lie between –1 and +1.
Q5. Are trigonometric ratios used in real life?
Yes, they are used in construction, navigation, physics, satellite systems, and engineering.
Q6. Are they important for competitive exams?
Absolutely, they are essential for Class 10–12 boards, JEE, CUET, NDA, and SSC exams.