Ever wondered how you can find the height of a tree without climbing it? Trigonometry makes that possible.
Trigonometric ratios tell us how the sides of a right‑angled triangle relate to its angles. Using those ratios, we can turn a tricky geometry problem into a quick calculation.
What are Trigonometric Ratios?
In a right‑angled triangle, the hypotenuse is the longest side opposite the right angle. The other two sides are called the adjacent (next to the angle we’re looking at) and the opposite (across from that angle).
The three basic ratios are:
- Sine (sin) = opposite ÷ hypotenuse
- Cosine (cos) = adjacent ÷ hypotenuse
- Tangent (tan) = opposite ÷ adjacent
Think of water flowing through a pipe: the amount that reaches the end (hypotenuse) depends on how much is fed in (opposite or adjacent). The ratios tell us the proportion.
Quick Example
Suppose a ladder leans against a wall, forming a 30° angle with the ground. The ladder is 5 m long (hypotenuse). What’s the height it reaches?
Using sin 30° = 0.5, height = sin 30° × hypotenuse = 0.5 × 5 m = 2.5 m.
Common Trigonometric Identities
Identities are equations that hold true for all angles. They let us swap one ratio for another, simplify expressions, and solve equations.
- Pythagorean identity: sin²θ + cos²θ = 1. (Here sin²θ means (sin θ)².)
- Tangent‑secant identity: 1 + tan²θ = sec²θ. (sec is the reciprocal of cos.)
- Cotangent‑cosecant identity: 1 + cot²θ = csc²θ. (cot is the reciprocal of tan; csc is the reciprocal of sin.)
These look fancy, but they’re just the algebraic versions of the Pythagoras theorem applied to a right‑angled triangle.
Worked Example – Using Identities
Find the value of sin θ if tan θ = 3/4 and θ is acute.
Step 1: Draw a right‑angled triangle where opposite = 3, adjacent = 4.
Step 2: Compute hypotenuse using Pythagoras: √(3²+4²)=5.
Step 3: sin θ = opposite ÷ hypotenuse = 3/5 = 0.6.
Notice how the identity sin²θ + cos²θ = 1 would give the same result if we first found cos θ = 4/5.
How to Use Trigonometric Ratios in Exams
CBSE questions love quick calculations. Keep these tricks handy:
- Memorise the values of sin, cos, tan for 0°, 30°, 45°, 60°, and 90°.
- When a problem gives a side length, write the ratio first, then plug numbers.
- Use identities to change a difficult expression into a known one.
- Check the quadrant (which part of the circle the angle lies in) to decide the sign (+ or –).
Mini Revision Table
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | Undefined |
Common Mistakes to Avoid
1. Mixing up opposite and adjacent sides for the given angle.
2. Forgetting that tan 90° is undefined (you can’t divide by zero).
3. Ignoring the sign rule when the angle is beyond 90°.
📝 Likely Exam Questions
- Find the value of cos θ if sin θ = 3/5 and θ is acute.
Answer: Using sin²θ + cos²θ = 1 → (3/5)² + cos²θ = 1 → 9/25 + cos²θ = 1 → cos²θ = 16/25 → cos θ = 4/5. - Calculate the height of a pole if the angle of elevation from a point 12 m away is 40°.
Answer: Height = 12 tan 40° ≈ 12 × 0.8391 ≈ 10.07 m. - Simplify: (1 + tan²θ) ÷ sec²θ.
Answer: Using identity 1 + tan²θ = sec²θ, the expression becomes sec²θ ÷ sec²θ = 1. - If the sides of a right‑angled triangle are in the ratio 5:12:13, find sin of the angle opposite the side 5.
Answer: sin θ = opposite ÷ hypotenuse = 5/13. - Prove that sin (90° – θ) = cos θ.
Answer: In a right‑angled triangle, the complement of an angle swaps the opposite and adjacent sides, so sin of the complement equals cos of the original.