ISC Class 11 Maths: Trigonometric Functions & Identities Explained

Ever wondered why the height of a roller‑coaster dip can be found with a simple ‘sin’ and a ruler? That’s trigonometry sneaking into everyday fun.

💡 In Simple Words: Trigonometric functions turn angles into numbers. They let us link the shape of a triangle to lengths we can measure, and identities are clever shortcuts that make those numbers easier to juggle.

What are Trigonometric Functions?

Imagine a right‑angled triangle drawn on a piece of paper. The angle you’re curious about sits at one corner. The sine (pronounced "sine") of that angle is the ratio of the side opposite the angle to the hypotenuse (the longest side). The cosine does the same but uses the side next to the angle. The tangent compares the opposite side to the adjacent side.

Hard term alert: hypotenuse – the side opposite the right angle, the longest side of the triangle. Once you picture a triangle, these ratios feel like recipes: mix the right sides, get a number that tells you something about the angle.

Why does this matter? Because the same ratios appear when you draw a circle of radius 1 (called the unit circle). The sine of an angle is just the y‑coordinate of a point on that circle, and the cosine is the x‑coordinate. Think of the unit circle as a giant clock face where the hand’s tip gives you the sine and cosine instantly.

Core Trigonometric Identities

Identities are equations that are always true, no matter which angle you plug in. They’re the cheat‑codes of trigonometry.

Pythagorean Identity

From the Pythagorean theorem (a² + b² = c²) you get sin²θ + cos²θ = 1. Here the superscript 2 means “squared” – multiply the number by itself. This identity is the backbone; it links sine and cosine forever.

Reciprocal Identities

These flip the ratios upside down. cscθ = 1/sinθ, secθ = 1/cosθ, and cotθ = 1/tanθ. The new names (csc, sec, cot) stand for cosecant, secant, and cotangent – just fancy ways to say “the reciprocal of sine, cosine, or tangent.”

Quotient Identities

Two of the original ratios can be written as a division of the other two: tanθ = sinθ / cosθ and cotθ = cosθ / sinθ. It’s like saying the height of a hill (tan) is the rise over the run (sin over cos).

Co‑function Identities

When you swap an angle with its complement (that is, 90° − θ), sine becomes cosine and vice‑versa: sin(90° − θ) = cosθ, cos(90° − θ) = sinθ. Imagine a seesaw: as one side goes up, the other goes down – they’re mirror images.

Worked Example: Simplify Using Identities

Problem: Simplify sin²θ + cos²θ − 2 sinθ cosθ.

Step 1: Spot the Pythagorean identity. The first two terms together equal 1.

Step 2: The remaining part is −2 sinθ cosθ. Recognise the double‑angle identity: sin2θ = 2 sinθ cosθ. So −2 sinθ cosθ = −sin2θ.

Result: The expression becomes 1 − sin2θ. That’s as simple as it gets.

Quick Reference Table

Tips to Remember Identities for the ISC Exam

• Write down the Pythagorean identity first – it’s your safety net.
• Group terms that look like a double‑angle pattern (2 sinθ cosθ or cos²θ − sin²θ).
• When an expression has both sine and cosine in the denominator, try converting to tan or cot using quotient identities.
• Check if the angle is complementary (90° − θ); if so, swap sin ↔ cos with co‑function identities.

📝 Likely Exam Questions

1. Prove that tan²θ + 1 = sec²θ.
   Model answer: Start with the Pythagorean identity, divide every term by cos²θ, obtain tan²θ + 1 = sec²θ.
2. Simplify cosθ · secθ.
   Model answer: secθ = 1/cosθ, so cosθ · (1/cosθ) = 1.
3. If sinθ = 3/5 and θ is acute, find cosθ and tanθ.
   Model answer: Use sin²θ + cos²θ = 1 → cosθ = 4/5. Then tanθ = sinθ/cosθ = (3/5)/(4/5) = 3/4.
4. Express sin2θ in terms of tanθ.
   Model answer: sin2θ = 2 sinθ cosθ = 2 tanθ/(1+tan²θ).
5. Show that sin(45°+θ) = (sinθ+cosθ)/√2.
   Model answer: Use the sine addition formula sin(A+B)=sinAcosB+cosAsinB with A=45°, sin45°=cos45°=√2/2.