What are the six trigonometric ratios?

sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent, and their reciprocals cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A.

Do trigonometric ratios change if the triangle's size changes?

No. As long as the angle stays the same, the ratio stays the same — any two right triangles sharing that angle are similar (AA criterion), so their corresponding side ratios are always equal.

How do I find all six trigonometric ratios if only one is given?

Write the two sides involved in the given ratio as k-multiples (e.g. if tan A = 4/3, let the opposite side be 4k and adjacent be 3k), find the third side using the Pythagoras theorem, then compute the remaining ratios directly.

What are the three trigonometric (Pythagorean) identities?

sin²A + cos²A = 1, 1 + tan²A = sec²A, and cot²A + 1 = cosec²A. All three come from dividing the Pythagoras equation AB²+BC²=AC² by AC², AB², and BC² respectively.

What are the standard trigonometric values for 0°, 30°, 45°, 60°, and 90°?

sin values are 0, 1/2, 1/√2, √3/2, 1 respectively. cos values are the exact same list in reverse: 1, √3/2, 1/√2, 1/2, 0.

Chapter 8 · CBSE Class 10 Maths

Introduction to Trigonometry

Two ways to learn — a crisp formula reference, or a story that makes the identities feel obvious.

Six trig ratiosStandard anglesPythagorean identitiesRatio invariance

Also useful:Chapter 7 — Coordinate Geometry Chapter 9 — Some Applications of Trigonometry NCERT Solutions — Class 10 Maths 1-on-1 CBSE Class 10 Tuition

CBSEClass 10MathematicsChapter 8⏱ 8 min crisp · 12 min story

Four scenes that build the ideas from scratch, step by step.

How this works

Four scenes, one thread: a trigonometric ratio is a property of an ANGLE, not of any particular triangle — which is why knowing just one ratio unlocks all six, and why three identities (built from Pythagoras) let every ratio be rewritten in terms of any other. Read the story, then tap "The Maths" for the formal result.

Same Angle, Same Ratio

Draw right triangle CAB with angle A at one corner. Now pick any point P further along the hypotenuse, and drop a perpendicular from P to form a SMALLER right triangle PAM, sharing the exact same angle A. Does sin A change in this smaller triangle?

Student

The triangle is smaller, so surely the side lengths — and therefore the ratio — must be different?

Look closer: △PAM and △CAB share angle A, and both have a right angle — that's two matching angles, which is exactly the AA similarity criterion from Chapter 6. Similar triangles have proportional corresponding sides.

△PAM ~ △CAB (AA similarity)

MP/AP = BC/AC (proportional sides)

But MP/AP is sin A in the small triangle, and BC/AC is sin A in the big triangle

Since those two ratios are forced to be equal by similarity, sin A comes out IDENTICAL in both triangles — no matter how small or large the triangle is drawn, as long as the angle stays the same.

One Ratio Unlocks Them All

Suppose all you're told is tan A = 4/3 — nothing about actual side lengths. Can the other five ratios (sin, cos, cosec, sec, cot) be found from just this?

Teacher

tan A = opposite/adjacent = 4/3 doesn't mean the sides ARE 4 and 3 — it means they're in that ratio. Call them 4k and 3k for some positive number k, then use Pythagoras to find the missing side.

opposite = 4k, adjacent = 3k

hypotenuse² = (4k)² + (3k)² = 25k² → hypotenuse = 5k

sin A = 4k/5k = 4/5, cos A = 3k/5k = 3/5

The remaining three are just reciprocals: cosec A = 5/4, sec A = 5/3, cot A = 3/4. The variable k cancelled out completely — which makes sense, since Scene 1 already showed ratios don't depend on actual size.

This "assume the sides are k times the ratio, then apply Pythagoras" move works for ANY single given ratio — sin, cos, or tan — always producing all six values.

The Universal Angles

Five particular angles — 0°, 30°, 45°, 60°, 90° — show up constantly, because they come from two special triangles: the 45-45-90 (an isosceles right triangle) and the 30-60-90 (half of an equilateral triangle).

Student

Do I need to re-derive these every time, or is there a pattern worth just memorising?

sin: 0°→0, 30°→1/2, 45°→1/√2, 60°→√3/2, 90°→1

cos: exactly the same list, reversed: 1, √3/2, 1/√2, 1/2, 0

Notice cos is just sin's table read backwards — that's not a coincidence (it previews the complementary-angle relationship explored further in this unit). tan follows directly since tan = sin/cos, and is undefined at 90° (dividing by cos 90° = 0).

One Identity, Many Disguises

Start from the one fact every right triangle already obeys: AB² + BC² = AC² (Pythagoras). Divide every term by AC².

Teacher

(AB/AC)² is just cos²A, and (BC/AC)² is sin²A. What does the divided equation say?

AB²/AC² + BC²/AC² = AC²/AC²

cos²A + sin²A = 1

Divide the SAME original equation by AB² instead, and a second identity falls out. Divide by BC² instead, and a third appears.

÷ AB²: 1 + tan²A = sec²A

÷ BC²: cot²A + 1 = cosec²A

All three identities are really the same Pythagoras fact, just divided differently. That means any trigonometric expression can be rewritten using whichever identity is most convenient.

Real use: to prove sec A(1−sin A)(sec A+tan A) = 1, rewrite everything in terms of sin A and cos A first — sec A(1−sinA)(secA+tanA) = (1/cosA)(1−sinA)((1+sinA)/cosA) = (1−sin²A)/cos²A = cos²A/cos²A = 1, using exactly the first identity in disguise.

The Big Picture

Every idea in this chapter branches from one fact: trigonometric ratios belong to the ANGLE, not the triangle. That single fact justifies finding all six ratios from just one, memorising a small table of standard angles, and using three Pythagoras-derived identities to rewrite any expression.

Does triangle size affect sin A?

No — only the angle matters (AA similarity)

Know one ratio, want all six?

Write sides as k-multiples, use Pythagoras

Standard angle values?

sin: 0,1/2,1/√2,√3/2,1 — cos is the same, reversed

The three core identities?

sin²+cos²=1, 1+tan²=sec², cot²+1=cosec²

Where do the identities come from?

Pythagoras, divided by AC², AB², BC² respectively

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