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Trigonometry

Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Define all six trigonometric ratios and move fluently between them.
2. Use standard angle values (0°, 30°, 45°, 60°, 90°) without hesitation.
3. Apply the three fundamental identities to simplify and prove expressions.
4. Handle complementary angles, compound angles, and multiple-angle formulas.
5. Solve heights-and-distances problems — the single most exam-favourite application.
6. Find maximum/minimum values of expressions like a·sin θ + b·cos θ — a classic Advanced-level trap zone.

The Hook: Measuring the Unmeasurable

In 1856, surveyors of the Great Trigonometrical Survey of India stood hundreds of kilometres away from a distant Himalayan peak. They never climbed it. They never touched it. Using only angles measured from the ground and the mathematics you are about to learn, they calculated its height — and declared it the tallest mountain on Earth. We now call it Mount Everest.

That is the power of trigonometry: it converts angles into distances. No ladder, no rope, no rocket — just a triangle and a few ratios. And that is exactly why examiners love it. One small formula, endless disguises.

"But Trigonometry Feels Like Memorising a Phone Directory…"

If that's you, you're not alone. Most aspirants struggle with trigonometry for one reason: they try to memorise it instead of seeing it. Every formula in this chapter is just a sentence about a right-angled triangle, translated into symbols. Once you see the triangle, the formula writes itself.

So let's rebuild trigonometry from one picture — a right-angled triangle with angle θ.

Chunk 1: The Six Ratios — One Triangle, Six Stories

In a right-angled triangle, for angle θ:

• Perpendicular (P) = side opposite to θ
• Base (B) = side adjacent to θ
• Hypotenuse (H) = side opposite the right angle (always the longest)

The six ratios:

Only three are fundamental. The other three (cosec, sec, cot) are just reciprocals. If you know sin, cos, tan — you know everything.

🧠 Active Recall: Close the book. Write the definitions of sin θ, cos θ, tan θ from memory. Now write sec θ in terms of cos θ. Check yourself.

Chunk 2: The Standard Angle Table — Derive, Don't Memorise

The famous table (0°, 30°, 45°, 60°, 90°) terrifies students. Here's the secret: you only need to memorise the sin row.

The Trick — "Square Root Ladder":
Write 0, 1, 2, 3, 4 under the five angles. Divide each by 4. Take the square root. Done.

• sin 0° = √(0/4) = 0
• sin 30° = √(1/4) = 1/2
• sin 45° = √(2/4) = 1/√2
• sin 60° = √(3/4) = √3/2
• sin 90° = √(4/4) = 1

Now:

• cos row = sin row written in reverse order.
• tan row = sin ÷ cos.

Three rows from one ladder. That's it.

🧠 Active Recall: Using the ladder, find sin 60°, cos 30°, and tan 45°. Notice anything about the first two?

Chunk 3: The Three Identities — The Constitution of Trigonometry

Everything in advanced trigonometry rests on one fact: Pythagoras' theorem, P² + B² = H². Divide this equation by H², B², and P² respectively, and you get the three fundamental identities:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = cosec²θ

Mnemonic — "Some Children Take Sweets":

• Sin² + Cos² = 1 (the parent identity)
• Tan goes with Sec (T–S pair)
• Cot goes with Cosec (the remaining pair)

Advanced application: Examiners rarely ask these directly. They hide them. Example: simplify (sin θ)/(1 + cos θ) + (1 + cos θ)/(sin θ). Take the LCM — the numerator becomes sin²θ + (1 + cos θ)² = sin²θ + 1 + 2cos θ + cos²θ = 2 + 2cos θ = 2(1 + cos θ). The answer collapses to 2/sin θ = 2 cosec θ. The identity did all the work.

🧠 Active Recall: From identity 1, derive identity 2 by dividing by cos²θ. Do it on paper now.

Chunk 4: Complementary Angles — The 90° Shortcut

When two angles add to 90°, their ratios swap partners:

• sin(90° − θ) = cos θ
• tan(90° − θ) = cot θ
• sec(90° − θ) = cosec θ

Memory hook: "Co" means "complement." Sine's complement is cosine. Tangent's is cotangent. Secant's is cosecant.

Classic exam pattern: "If sin 3A = cos(A − 26°), find A." Since sin x = cos(90° − x), we get 3A + (A − 26°) = 90°, so 4A = 116°, A = 29°. One identity, ten seconds.

🧠 Active Recall: What is tan 1° · tan 2° · tan 3° … tan 89°? (Hint: pair tan 1° with tan 89° = cot 1°.)

Chunk 5: Compound & Multiple Angles — The Advanced Arsenal

These formulas power the toughest questions in SSC CGL Tier-II and IBPS PO Mains:

Compound angles:

• sin(A + B) = sin A cos B + cos A sin B
• cos(A + B) = cos A cos B − sin A sin B
• tan(A + B) = (tan A + tan B) / (1 − tan A tan B)

Double angles (put B = A above):

• sin 2A = 2 sin A cos A
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
• tan 2A = 2 tan A / (1 − tan²A)

The examiner's favourite — the max-min rule:
For any expression a sin θ + b cos θ, the value always lies between −√(a² + b²) and +√(a² + b²).

Example: maximum value of 3 sin θ + 4 cos θ? Answer: √(9 + 16) = 5. (Recognise the 3-4-5 triangle? That's no coincidence.)

🧠 Active Recall: What is the minimum value of 5 sin θ + 12 cos θ? And what is sin 2A when sin A = 3/5, cos A = 4/5?

Chunk 6: Heights & Distances — Where Trigonometry Earns Its Salary

This is the most frequently tested application. The whole game is three steps:

1. Draw the picture. Observer, object, ground — a right triangle is always hiding there.
2. Identify the angle. Angle of elevation (looking up) or depression (looking down — equal to the elevation from the other side, by alternate angles).
3. Pick the ratio. Almost always tan θ = height / distance, because height and ground distance are the two sides you deal with.

The two golden values:

• tan 45° = 1 → height = distance
• tan 30° = 1/√3 and tan 60° = √3 → these two angles appear in the majority of problems, often in the same question (a tower viewed from two points).

Classic pattern: From a point, the angle of elevation of a tower's top is 30°. Walking 20 m closer, it becomes 60°. Height? Set up h = d·tan 60° and h = (d + 20)·tan 30°. Solving gives h = 10√3 m. Learn this template — it repeats endlessly.

🧠 Active Recall: The shadow of a pole equals its height. What is the sun's angle of elevation? (No calculator needed.)

Chunk 7: Signs in the Four Quadrants (Advanced Extension)

For angles beyond 90°, use the unit circle. The sign rule:

"All Silver Tea Cups" (or "Add Sugar To Coffee") — going anticlockwise from the first quadrant:

So sin 150° = sin(180° − 30°) = +sin 30° = 1/2, but cos 150° = −√3/2.

🧠 Active Recall: Is tan 225° positive or negative? What is its value?

Summary Table

🔁 Review This After 2 Days

Don't re-read passively. Instead, take a blank page and reconstruct from memory: (1) the standard angle table using the ladder, (2) the three identities, (3) the max-min rule, and (4) one heights-and-distances template. Whatever you can't reproduce — that's your revision list. Trigonometry is not memorised; it is rebuilt, and every rebuild makes it faster.