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Class 10 · Math

Introduction to Trigonometry

Trigonometry is the mathematics of triangles and angles.

Feynman Lens

Start with the simplest version: this lesson is about Introduction to Trigonometry. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.

Trigonometry is the mathematics of triangles and angles. It reveals profound relationships between angles and side lengths that allow you to find invisible measurements—the height of mountains, the distance to stars, or the depth of oceans. Rather than using geometry tools, you use ratios that depend only on the angle. This chapter introduces trigonometric ratios (sine, cosine, tangent) in right triangles, opening doors to physics, engineering, navigation, and astronomy. Trigonometry transforms difficult geometric problems into elegant algebraic relationships.

Right Triangles: The Foundation

All trigonometry in this chapter begins with right triangles—triangles with one 90° angle.

Labeling convention:

Key insight: In a right triangle, once you know one acute angle, the shape is determined (by chapter-06-triangles similarity). Different right triangles with the same angle are all similar, so the ratios of sides remain constant!

The Six Trigonometric Ratios

For angle A in a right triangle, the ratios are:

sin(A) = opposite/hypotenuse
cos(A) = adjacent/hypotenuse
tan(A) = opposite/adjacent

csc(A) = hypotenuse/opposite (reciprocal of sine)
sec(A) = hypotenuse/adjacent (reciprocal of cosine)
cot(A) = adjacent/opposite (reciprocal of tangent)

Memory aid: SOH-CAH-TOA

Critical property: These ratios depend ONLY on the angle, not on the size of the triangle. This is why you can calculate hidden measurements using just an angle measurement.

Trigonometric Ratios for Standard Angles

The ratios for 0°, 30°, 45°, 60°, and 90° appear so frequently they're worth memorizing:

Angle  |  sin  |  cos  |  tan
-------|-------|-------|-------
  0°   |   0   |   1   |   0
 30°   |  1/2  | √3/2  | 1/√3
 45°   | 1/√2  | 1/√2  |   1
 60°   | √3/2  | 1/2   |  √3
 90°   |   1   |   0   |  ∞

These values appear in problems so often that memorizing them saves time and prevents calculation errors.

Trigonometric Identities: Fundamental Relationships

Pythagorean Identity (from the Pythagorean theorem):

sin²(A) + cos²(A) = 1

This identity connects sine and cosine, allowing you to find one if you know the other.

Complementary Angle Relationships:

sin(A) = cos(90° − A)
cos(A) = sin(90° − A)
tan(A) = cot(90° − A)

Two angles that sum to 90° are complementary. Their trigonometric ratios are swapped in this specific way.

Real-World Application: Heights and Angles of Elevation

The scenario: You want to find the height of a building. You know:

Solution: Using tan(angle) = opposite/adjacent:

This simple formula converts an angle measurement into a height—without needing to actually climb!

Example: Standing 50 meters from a building, you measure an elevation angle of 60°.

Trigonometry builds naturally into:

Key Ratios and Identities

Socratic Questions

  1. Why do trigonometric ratios depend only on the angle, not on the size of the triangle? What property of triangles ensures this?
  1. If sin(30°) = 1/2, what is sin(60°)? Use the complementary angle relationship to explain your answer.
  1. In a right triangle, if sin(A) = 3/5, can you find cos(A) without finding the actual angle A? How?
  1. Why is tan(45°) = 1? What does this tell you about a 45°-45°-90° triangle's shape?
  1. If you're standing closer to a building, does the angle of elevation to its roof increase or decrease? Why does this make intuitive sense from the tangent ratio?
🃏 Flashcards — Quick Recall
Term / Concept
What is Introduction to Trigonometry?
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Introduction to Trigonometry is the central idea of this lesson. Use the chapter examples to explain what it means and why it matters.
Term / Concept
What is Labeling convention?
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- The right angle is at C
Term / Concept
What is Key insight?
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In a right triangle, once you know one acute angle, the shape is determined (by chapter-06-triangles similarity). Different right triangles with the same angle are all similar, so the ratios of sides remain constant!
Term / Concept
What is Critical property?
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These ratios depend ONLY on the angle, not on the size of the triangle. This is why you can calculate hidden measurements using just an angle measurement.
Term / Concept
What is Pythagorean Identity?
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(from the Pythagorean theorem):
Term / Concept
What is The scenario?
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You want to find the height of a building. You know:
Term / Concept
What is Solution?
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Using tan(angle) = opposite/adjacent:
Term / Concept
What is Example?
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Standing 50 meters from a building, you measure an elevation angle of 60°.
Term / Concept
What is sin(A) = opposite/hypotenuse?
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- cos(A) = adjacent/hypotenuse
Term / Concept
What is tan(A) = opposite/adjacent = sin(A)/cos(A)?
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- sin²(A) + cos²(A) = 1 (Pythagorean Identity)
Term / Concept
What is tan(A) = 1/cot(A)?
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- sin(A) = cos(90° − A) (Complementary angles)
Term / Concept
What is the core idea of Right Triangles: The Foundation?
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All trigonometry in this chapter begins with right triangles—triangles with one 90° angle.
Term / Concept
What is the core idea of The Six Trigonometric Ratios?
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For angle A in a right triangle, the ratios are: sin(A) = opposite/hypotenuse cos(A) = adjacent/hypotenuse tan(A) = opposite/adjacent csc(A) = hypotenuse/opposite (reciprocal of sine) sec(A) = hypotenuse/adjacent…
Term / Concept
What is the core idea of Trigonometric Ratios for Standard Angles?
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The ratios for 0°, 30°, 45°, 60°, and 90° appear so frequently they're worth memorizing: Angle | sin | cos | tan 0° | 0 | 1 | 0 30° | 1/2 | √3/2 | 1/√3 45° | 1/√2 | 1/√2 | 1 60° | √3/2 | 1/2 | √3 90° | 1 | 0 | ∞ These…
Term / Concept
What is the core idea of Trigonometric Identities: Fundamental Relationships?
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Pythagorean Identity (from the Pythagorean theorem): sin²(A) + cos²(A) = 1 This identity connects sine and cosine, allowing you to find one if you know the other.
Term / Concept
What is the core idea of Real-World Application: Heights and Angles of Elevation?
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The scenario: You want to find the height of a building.
Term / Concept
What is the core idea of Connecting to Related Topics?
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Trigonometry builds naturally into: - chapter-06-triangles: Similar triangles justify why trig ratios depend only on angles - chapter-07-coordinate-geometry: Angles in coordinate geometry use trigonometric concepts -…
Term / Concept
What is the core idea of Key Ratios and Identities?
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- sin(A) = opposite/hypotenuse - cos(A) = adjacent/hypotenuse - tan(A) = opposite/adjacent = sin(A)/cos(A) - sin²(A) + cos²(A) = 1 (Pythagorean Identity) - tan(A) = 1/cot(A) - sin(A) = cos(90° − A) (Complementary angles)
Term / Concept
What is The right angle is at C?
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The right angle is at C
Term / Concept
What is The angle of interest (say, A) is?
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The angle of interest (say, A) is at a vertex
Term / Concept
What is The side opposite to angle A is?
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The side opposite to angle A is labeled 'a' (or 'opposite')
Term / Concept
What is The side adjacent to angle A is?
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The side adjacent to angle A is labeled 'b' (or 'adjacent')
Term / Concept
What is The side opposite the right angle is?
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The side opposite the right angle is the hypotenuse (labeled 'h')
Term / Concept
What is Sine = Opposite/Hypotenuse?
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Sine = Opposite/Hypotenuse
Term / Concept
What is Cosine = Adjacent/Hypotenuse?
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Cosine = Adjacent/Hypotenuse
Term / Concept
What is Tangent = Opposite/Adjacent?
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Tangent = Opposite/Adjacent
Term / Concept
What is Your distance from the building (adjacent side)?
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Your distance from the building (adjacent side)
Term / Concept
What is The angle of elevation (angle from horizontal?
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The angle of elevation (angle from horizontal to your line of sight)
Term / Concept
What is height = distance × tan(elevation angle)?
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height = distance × tan(elevation angle)
Term / Concept
What is Height = 50 × tan(60°) = 50?
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Height = 50 × tan(60°) = 50 × √3 ≈ 86.6 meters
Term / Concept
What is chapter-06-triangles?
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Similar triangles justify why trig ratios depend only on angles
Term / Concept
What is chapter-07-coordinate-geometry?
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Angles in coordinate geometry use trigonometric concepts
Term / Concept
What is chapter-09-some-applications-of-trigonometry?
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Real-world applications of angle and distance problems
Term / Concept
What is chapter-10-circles?
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Angles and radii in circles use trigonometric relationships
Term / Concept
What is cos(A) = adjacent/hypotenuse?
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cos(A) = adjacent/hypotenuse
Term / Concept
What is sin²(A) + cos²(A) = 1 (Pythagorean Identity)?
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sin²(A) + cos²(A) = 1 (Pythagorean Identity)
Term / Concept
What is sin(A) = cos(90° − A) (Complementary angles)?
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sin(A) = cos(90° − A) (Complementary angles)
Term / Concept
What does this formula or relation mean: sin(A) = opposite/hypotenuse cos(A) = adjacent/h?
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This formula or symbolic relation appears in the chapter. Explain what each part represents before using it.
Term / Concept
What does this formula or relation mean: Angle | sin | cos | tan -------|-------|-------|?
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This formula or symbolic relation appears in the chapter. Explain what each part represents before using it.
Term / Concept
What does this formula or relation mean: sin²(A) + cos²(A) = 1?
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This formula or symbolic relation appears in the chapter. Explain what each part represents before using it.
40 cards — click any card to flip
📝 Quick Quiz — Test Yourself
Why do trigonometric ratios depend only on the angle, not on the size of the triangle? What property of triangles ensures this?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
If sin(30°) = 1/2, what is sin(60°)? Use the complementary angle relationship to explain your answer.
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
In a right triangle, if sin(A) = 3/5, can you find cos(A) without finding the actual angle A? How?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Why is tan(45°) = 1? What does this tell you about a 45°-45°-90° triangle's shape?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
If you're standing closer to a building, does the angle of elevation to its roof increase or decrease? Why does this make intuitive sense from the tangent ratio?
  • A Memorize the exact line without checking the reasoning.
  • B Use the chapter's formula or relation and explain the reasoning step by step.
  • C Ignore the examples and rely only on a keyword.
  • D Treat the idea as unrelated to the rest of the lesson.
Which approach best shows that you understand Introduction to Trigonometry?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Labeling convention?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Key insight?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Critical property?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Pythagorean Identity?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The scenario?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Solution?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Example?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand sin(A) = opposite/hypotenuse?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand tan(A) = opposite/adjacent = sin(A)/cos(A)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand tan(A) = 1/cot(A)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Right Triangles: The Foundation?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The Six Trigonometric Ratios?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Trigonometric Ratios for Standard Angles?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Trigonometric Identities: Fundamental Relationships?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Real-World Application: Heights and Angles of Elevation?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Connecting to Related Topics?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Key Ratios and Identities?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The right angle is at C?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The angle of interest (say, A) is?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The side opposite to angle A is?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The side adjacent to angle A is?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The side opposite the right angle is?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Sine = Opposite/Hypotenuse?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Cosine = Adjacent/Hypotenuse?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Tangent = Opposite/Adjacent?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Your distance from the building (adjacent side)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand The angle of elevation (angle from horizontal?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand height = distance × tan(elevation angle)?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand Height = 50 × tan(60°) = 50?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-06-triangles?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-07-coordinate-geometry?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-09-some-applications-of-trigonometry?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand chapter-10-circles?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
Which approach best shows that you understand cos(A) = adjacent/hypotenuse?
  • A Repeat its name from memory.
  • B Explain it using a simple example and the reason it works.
  • C Skip the conditions where it applies.
  • D Use it only when the textbook wording is identical.
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