Schooling · Class 10 · Mathematics · Introduction to Trigonometry · Chapter on a page
CBSE Class 10 · Mathematics · NCERT Mathematics
Introduction to Trigonometry: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- In a right-angled triangle, for a…
- The three primary ratios are
- The three reciprocal ratios are
- Standard angle values must be…
- Complementary-angle relations
- Three fundamental identities hold…
- Formulas and facts
- sin θ = opposite / hypotenuse
- cos θ = adjacent / hypotenuse
- Formula
- cosec θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ = cos θ / sin θ
- sin²θ + cos²θ = 1
- Watch out for
- Confusing opposite and adjacent sides
- Writing sin 60° = 60 or treating sin…
- Using tan 90° as a finite number
- Forgetting to square correctly in…
- Mixing up complementary relations
What this chapter is about
Trigonometry means "measurement of triangles." This chapter introduces you to a branch of mathematics that connects angles with ratios of sides in a right-angled triangle. You will learn how the size of an acute angle in a right triangle determines fixed ratios between pairs of sides, no matter how large or small the triangle is.
Key ideas
- In a right-angled triangle, for a given acute angle, the ratio of any two sides stays the same regardless of the triangle's size; this constant ratio is called a trigonometric ratio.
- The three primary ratios are: sin θ = (side opposite to θ) / hypotenuse, cos θ = (side adjacent to θ) / hypotenuse, tan θ = (side opposite) / (side adjacent).
- The three reciprocal ratios are: cosec θ = 1 / sin θ, sec θ = 1 / cos θ, cot θ = 1 / tan θ.
- Standard angle values must be memorised: for example, sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1, sin 90° = 1, cos 0° = 1.
- Complementary-angle relations: sin (90° − θ) = cos θ, cos (90° − θ) = sin θ, tan (90° − θ) = cot θ, and similarly for the reciprocal ratios.
- Three fundamental identities hold for every acute angle θ: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ.
Formulas and facts to remember
- Formula: sin θ = opposite / hypotenuse · Meaning: Ratio of the side facing the angle to the longest side.
- Formula: cos θ = adjacent / hypotenuse · Meaning: Ratio of the side next to the angle to the longest side.
- Formula: tan θ = opposite / adjacent = sin θ / cos θ · Meaning: Ratio of the side facing the angle to the side next to it.
- Formula: cosec θ = 1 / sin θ · Meaning: Reciprocal of sine.
- Formula: sec θ = 1 / cos θ · Meaning: Reciprocal of cosine.
- Formula: cot θ = 1 / tan θ = cos θ / sin θ · Meaning: Reciprocal of tangent.
- Formula: sin²θ + cos²θ = 1 · Meaning: Pythagorean identity linking sine and cosine.
- Formula: 1 + tan²θ = sec²θ · Meaning: Derived from the first identity by dividing by cos²θ.
- Formula: 1 + cot²θ = cosec²θ · Meaning: Derived from the first identity by dividing by sin²θ.
- Formula: sin (90° − θ) = cos θ, cos (90° − θ) = sin θ · Meaning: Complementary-angle relations.
- Standard values:
- θ: sin θ · 0°: 0 · 30°: 1/2 · 45°: 1/√2 · 60°: √3/2 · 90°: 1
- θ: cos θ · 0°: 1 · 30°: √3/2 · 45°: 1/√2 · 60°: 1/2 · 90°: 0
- θ: tan θ · 0°: 0 · 30°: 1/√3 · 45°: 1 · 60°: √3 · 90°: not defined
Common mistakes
- Confusing opposite and adjacent sides → always label sides with respect to the angle you are working with, not the right angle.
- Writing sin 60° = 60 or treating sin as a multiplier → sin is a ratio, not a quantity to multiply by the angle.
- Using tan 90° as a finite number → tan 90° is undefined because cos 90° = 0.
- Forgetting to square correctly in identities: writing sin θ + cos θ = 1 instead of sin²θ + cos²θ = 1 → the identity involves squares.
- Mixing up complementary relations: writing sin (90° − θ) = sin θ → it equals cos θ, not sin θ.
Quick revision
- sin²θ + cos²θ = 1 is the master identity; the other two follow from it.
- tan θ = sin θ / cos θ; cot θ = cos θ / sin θ.
- Memorise standard values for 0°, 30°, 45°, 60°, 90° — most problems use them.
- Complementary angles add to 90°; their sine and cosine swap.
- Always identify which side is opposite and which is adjacent before writing a ratio.