Introduction to Trigonometry: Decoding Right Triangles

🗼8.1 Right Triangles Hiding in Plain Sight

Picture three everyday moments: a student looking up at the top of the Qutub Minar, a girl on a balcony looking down at a flower pot across a river, and a mother spotting a hot air balloon rising in the sky. In every single one of these, if you draw a line of sight and a line straight down (or straight across) to the ground, you get a right triangle — even though nobody drew it on purpose.

The word trigonometry comes from three Greek words: tri (three), gon (sides), and metron (measure) — literally, "measuring three-sided figures." It's the study of the relationships between the sides and angles of a triangle. The earliest known work on it comes from ancient Egypt and Babylon, and early astronomers used it to calculate the distances of stars and planets from Earth.

🎯What This Chapter Covers

• Trigonometric ratios of an acute angle in a right triangle — sin, cos, tan, and their reciprocals.
• Exact values of these ratios for the specific angles 0°, 30°, 45°, 60°, and 90°.
• Trigonometric identities — equations relating these ratios that hold true for every acute angle.

📐8.2 Trigonometric Ratios: Naming the Sides

Take a right triangle ABC, right-angled at B, and focus on the acute angle A. The side BC directly faces angle A, so it's called the side opposite to A. The hypotenuse AC is the longest side, always across from the right angle. The remaining side AB, which forms part of angle A itself, is called the side adjacent to A.

🔢The Six Trigonometric Ratios

Using these three side names, we define six ratios for angle A. The first three are the ones you'll use constantly; the last three are simply their reciprocals.

📏Ratios Don't Care About the Triangle's Size

Here's the key insight that makes trigonometric ratios genuinely useful: if you pick any other point P on the hypotenuse AC and drop a perpendicular to AB, the smaller triangle you create is similar to the original triangle ABC (by the AA similarity criterion from Chapter 6, since both share angle A and both have a right angle). Because similar triangles have proportional sides, every ratio you compute for angle A comes out identical, no matter how big or small the triangle is.

✏️Worked Examples

🔺8.3 Trigonometric Ratios of Some Specific Angles

Some angles come up so often — 0°, 30°, 45°, 60°, 90° — that it's worth working out their exact trigonometric ratios once and for all, instead of measuring a triangle every time.

◣45° — From an Isosceles Right Triangle

In a right triangle ABC, right-angled at B, if one acute angle is 45°, the other acute angle must also be 45° (since the three angles sum to 180°). That forces BC = AB — call this common length a.

△30° and 60° — From an Equilateral Triangle

Take an equilateral triangle ABC, where every angle is 60°. Drop a perpendicular AD from A to BC. By congruence (△ABD ≅ △ACD), D is the midpoint of BC, and AD bisects ∠A into two 30° halves. This creates a right triangle ABD with ∠BAD = 30° and ∠ABD = 60°.

↔️0° and 90° — The Limiting Cases

Imagine shrinking angle A in a right triangle ABC closer and closer to 0°. As this happens, side BC (opposite to A) shrinks toward 0, while AC becomes nearly equal to AB. That gives us sin 0° = 0 and cos 0° = 1 — definitions, not computations, since a "triangle" with a 0° angle isn't really a triangle anymore.

Push angle A the other way, toward 90°, and the opposite happens: side AB shrinks toward 0 while AC becomes nearly equal to BC. That gives sin 90° = 1 and cos 90° = 0.

📊The Complete Reference Table

✏️Worked Examples

🔗8.4 Trigonometric Identities

An identity is an equation that's true for every value of the variable involved — not just some special cases. A trigonometric identity is one involving trigonometric ratios that holds for every value of the angle(s) involved. This section proves three such identities, all flowing from a single source: the Pythagoras theorem.

①The First Identity: sin²A + cos²A = 1

Start with the Pythagoras theorem in right triangle ABC (right-angled at B): AB² + BC² = AC². Now divide every single term by AC².

②The Second Identity: 1 + tan²A = sec²A

Take the very same Pythagoras equation, AB² + BC² = AC², but this time divide every term by AB² instead.

③The Third Identity: cot²A + 1 = cosec²A

One more division of the same Pythagoras equation, this time by BC².

These three identities are powerful because they let you find every other trigonometric ratio once you know just one. For instance, if tan A = 1/√3, then sec²A = 1 + 1/3 = 4/3, so sec A = 2/√3, which gives cos A = √3/2, and then sin A = √(1−cos²A) = 1/2, giving cosec A = 2.

✏️Worked Examples

📝8.5 Chapter Summary

1. In right triangle ABC, right-angled at B: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent.
2. Reciprocals: cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A. Also, tan A = sin A/cos A and cot A = cos A/sin A.
3. Knowing one ratio unlocks the rest — use the k-trick plus Pythagoras, or the identities directly.
4. Exact values exist for 0°, 30°, 45°, 60°, 90° — memorise the reference table.
5. Range facts: sin A and cos A never exceed 1. sec A (for 0°≤A<90°) and cosec A (for 0°<A≤90°) are always ≥ 1.
6. The three identities: sin²A + cos²A = 1 (all 0°–90°); sec²A − tan²A = 1 (0°≤A<90°); cosec²A − cot²A = 1 (0°<A≤90°).

TOOL 1 Six-Ratio Solver

Enter any one trigonometric ratio as a fraction (e.g. tan A = 4/3) and instantly see all six ratios worked out, using the same k-multiple + Pythagoras method as the chapter's worked examples.

TOOL 2 Special Angle Expression Evaluator

Pick an expression built from sin, cos, tan, cosec, sec and cot of 0°, 30°, 45°, 60° and 90°, and see it evaluated step by step using the reference table.