ISC Class 11 Mathematics Trigonometric Functions & Identities

Ever wondered why engineers talk about sine waves when they design bridges? That’s trigonometry sneaking into the real world.

Trigonometric functions are just ways to relate angles to side lengths in a right‑angled triangle. Identities are handy shortcuts that let you swap one trig expression for another, making calculations faster.

What are Trigonometric Functions?

Imagine a right‑angled triangle drawn on a piece of graph paper. The hypotenuse (the longest side) is like the diagonal of a rectangular screen, while the other two sides are its height and base. The three basic trigonometric functions—sine, cosine, and tangent—compare one side to another:

• Sine (sin) = opposite side ÷ hypotenuse.
• Cosine (cos) = adjacent side ÷ hypotenuse.
• Tangent (tan) = opposite side ÷ adjacent side.

Think of water flowing through a pipe. If the pipe’s width is the “adjacent” side and the water height is the “opposite” side, the ratio tells you how steep the flow is—exactly what tan measures.

Extending to the Unit Circle

When you move beyond right triangles, the unit circle (a circle with radius 1) helps. Any angle θ measured from the positive x‑axis lands at a point (x, y). Here, cos θ = x and sin θ = y. The tangent becomes tan θ = y/x as long as x ≠ 0. This view lets you work with angles bigger than 90° and even negative angles.

Key Trigonometric Ratios and Values

Memorising a few special angles saves a lot of time. For 0°, 30°, 45°, 60°, and 90° you can picture a 30‑60‑90 or a 45‑45‑90 triangle, or just read them off the unit circle.

Notice how the numbers flip around 45°—that’s the co‑function relationship, which we’ll see in the identities section.

Common Trigonometric Identities

Identities are equations that are always true, no matter what angle you plug in. They’re like the grammar rules of the trig language.

Pythagorean Identities

Derived from the Pythagorean theorem (a² + b² = c²) applied to the unit circle, the most famous one is:

sin²θ + cos²θ = 1

Here, the superscript 2 means “squared” (multiplied by itself). You can rearrange it to get tan²θ + 1 = sec²θ or 1 + cot²θ = csc²θ, where sec (secant) = 1/cos and csc (cosecant) = 1/sin. These are the reciprocal identities.

Reciprocal Identities

They simply say that each function has a partner that’s its “flip”. For example:

• csc θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ

Quotient Identities

These turn a ratio of sine and cosine into tangent or cotangent:

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

Co‑function Identities

They capture the idea that sine and cosine are just “shifted” versions of each other. The rule is:

sin(90° – θ) = cos θ and cos(90° – θ) = sin θ

It’s like swapping the height and base of a right triangle when you rotate it 90°.

Even‑Odd Identities

These tell you how a function behaves for negative angles:

• Even functions stay the same: cos(–θ) = cos θ.
• Odd functions change sign: sin(–θ) = –sin θ and tan(–θ) = –tan θ.

Double‑Angle Identities

They let you express trig values of twice an angle using the original angle. A popular one is:

sin 2θ = 2 sin θ cos θ

Another useful version for cosine is:

cos 2θ = cos²θ – sin²θ (or rewrite using the Pythagorean identity to get cos 2θ = 2 cos²θ – 1 or cos 2θ = 1 – 2 sin²θ).

Worked Example 1: Using the Pythagorean Identity

Find cos θ if sin θ = 3/5 and θ is an acute angle (0° 

1. Start with sin²θ + cos²θ = 1.
2. Square the given sine: (3/5)² = 9/25.
3. Plug in: 9/25 + cos²θ = 1.
4. Subtract 9/25 from both sides: cos²θ = 1 – 9/25 = 16/25.
5. Take the square root. Since θ is acute, cosine is positive, so cos θ = 4/5.

That’s it—one line of algebra and the identity did the heavy lifting.

Worked Example 2: Simplifying with Double‑Angle

Simplify sin 2θ / (1 + cos 2θ).

1. Replace sin 2θ with 2 sin θ cos θ (double‑angle identity).
2. Replace cos 2θ with 2 cos²θ – 1 (one form of the cosine double‑angle).
3. The expression becomes (2 sin θ cos θ) / [1 + (2 cos²θ – 1)].
4. Simplify the denominator: 1 + 2 cos²θ – 1 = 2 cos²θ.
5. Now we have (2 sin θ cos θ) / (2 cos²θ). Cancel the common factor 2 cos θ.
6. Result: sin θ / cos θ = tan θ.

So the original fraction just equals tan θ—a neat shortcut for exams.

Quick Reference Table

Bullet Summary

• Trigonometric functions relate angles to side ratios.
• Unit circle extends definitions to any angle.
• Memorise key values for 0°, 30°, 45°, 60°, 90°.
• Pythagorean identity is the backbone for many derivations.
• Reciprocal, quotient, co‑function, even‑odd, and double‑angle identities are the main tool‑kits for simplifying.
• Practice rewriting expressions using these identities; speed comes from pattern recognition.

📝 Likely Exam Questions

1. Find the exact value of cos θ if sin θ = 5/13 and θ is in the first quadrant.
   Answer: Using sin²θ + cos²θ = 1 → cos²θ = 1 – (5/13)² = 144/169 → cos θ = 12/13.
2. Simplify the expression (1 – cos 2θ) / sin 2θ.
   Answer: Replace 1 – cos 2θ with 2 sin²θ and sin 2θ with 2 sin θ cos θ → (2 sin²θ)/(2 sin θ cos θ) = tan θ.
3. Prove that tan θ + cot θ = sec θ csc θ.
   Answer: Write tan θ = sin θ/cos θ and cot θ = cos θ/sin θ. Their sum = (sin²θ + cos²θ)/(sin θ cos θ) = 1/(sin θ cos θ) = (1/cos θ)(1/sin θ) = sec θ csc θ.
4. If sin α = 3/5 and α is acute, compute sin 2α.
   Answer: Use sin 2α = 2 sin α cos α. First find cos α = 4/5. Then sin 2α = 2·(3/5)·(4/5) = 24/25.
5. Show that cos (π – θ) = –cos θ.
   Answer: Using the unit circle, the angle (π – θ) lands in the second quadrant where the x‑coordinate is the negative of the x‑coordinate at θ. Hence cos(π – θ) = –cos θ.