Trigonometry Class 10: Formulas & Identities Cheat Sheet

Q: What are the six trigonometric ratios?

The six trigonometric ratios of an acute angle in a right triangle are sine (sinθ = opposite/hypotenuse), cosine (cosθ = adjacent/hypotenuse), tangent (tanθ = opposite/adjacent), and their reciprocals cosecant (cosecθ = 1/sinθ), secant (secθ = 1/cosθ), and cotangent (cotθ = 1/tanθ).

Q: What are the three Pythagorean trigonometric identities?

The three identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. Each follows from the Pythagoras theorem applied to a right triangle with angle θ.

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

sin: 0, 1/2, 1/√2, √3/2, 1. cos: 1, √3/2, 1/√2, 1/2, 0. tan: 0, 1/√3, 1, √3, undefined (not defined at 90°). These five angles form the standard table used throughout the chapter.

Q: What is the relationship between trigonometric ratios of complementary angles?

For complementary angles (angles that add up to 90°): sin(90°−θ) = cosθ, cos(90°−θ) = sinθ, tan(90°−θ) = cotθ, cot(90°−θ) = tanθ, sec(90°−θ) = cosecθ, and cosec(90°−θ) = secθ.

Q: What are the most common mistakes in Class 10 Trigonometry?

Common mistakes include mixing up which side is opposite versus adjacent to the angle in question, writing a reciprocal ratio incorrectly (such as cosecθ = cosθ instead of 1/sinθ), misapplying the Pythagorean identities, using the complementary angle rule on angles that don't actually add to 90°, and treating trigonometric ratios as ordinary algebraic variables that can be cancelled freely.

Introduction to Trigonometry Class 10 with formulas, identities and sin cos tan

Introduction to Trigonometry — Class 10 Formulas & Identities Cheat Sheet (2026)

📅 Updated Sep 11, 2026 ⏱ 14 min read 📘 CBSE / NCERT Class 10, Chapter 8 ✍️ MyTestSeries Expert Team

Introduction to Trigonometry Class 10 — ratios, identities and standard angle table explained by MyTestSeries

Trigonometry can feel like a sudden shift after Coordinate Geometry — new symbols, a table of "special" angles to memorise, and identities that look intimidating at first glance. But underneath all of it is one simple picture: a right triangle, an angle, and three ratios between its sides. Everything else in this chapter — the reciprocal ratios, the identities, the complementary angle rules — is built directly on top of that one picture.

This cheat sheet lays out every ratio, the full standard angle table, all three Pythagorean identities, and the complementary angle relationships — each with a worked example — followed by the specific mistakes that quietly cost marks: side mix-ups, reciprocal errors, and misapplied identities.

6Trig Ratios

3Key Identities

6Solved Examples

7Common Mistakes

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1. The Six Trigonometric Ratios

Picture a right triangle with one acute angle θ. Relative to θ, the three sides are: the hypotenuse (opposite the right angle), the opposite side (opposite θ itself), and the adjacent side (next to θ, not the hypotenuse). Six ratios can be formed from these three sides:

sinθ = opposite/hypotenuse cosθ = adjacent/hypotenuse tanθ = opposite/adjacent
cosecθ = 1/sinθ secθ = 1/cosθ cotθ = 1/tanθ Also: tanθ = sinθ/cosθ, and cotθ = cosθ/sinθ — both follow directly from the definitions above.

💡 Quick check: Which side is "opposite" and which is "adjacent" depends entirely on which angle you're looking from — for the same triangle, the two acute angles swap opposite and adjacent sides between them.

2. Standard Angle Table (0°–90°)

θ	0°	30°	45°	60°	90°
sinθ	0	1/2	1/√2	√3/2	1
cosθ	1	√3/2	1/√2	1/2	0
tanθ	0	1/√3	1	√3	Not defined

The reciprocal ratios follow directly: cosecθ, secθ, and cotθ are just 1 divided by sinθ, cosθ, and tanθ respectively at each angle. Note cosecθ is undefined at 0°, and secθ is undefined at 90° — exactly where their reciprocal partner equals zero.

3. The Three Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ All three follow from applying the Pythagoras theorem to the same right triangle — divide through by hypotenuse², adjacent², or opposite² respectively to get each version.

Ratios, identities, and a full angle table to memorise — practice is what makes them automatic. Try the 7-Day Demo Test Series for instant, explained solutions after every attempt.

4. Complementary Angle Relationships

sin(90°−θ) = cosθ cos(90°−θ) = sinθ tan(90°−θ) = cotθ
cot(90°−θ) = tanθ sec(90°−θ) = cosecθ cosec(90°−θ) = secθ Applies only when the two angles genuinely add up to 90° — the "co-" in cosine, cosecant, and cotangent literally stands for "complement."

5. Six Fully Solved Examples

Example 1 — Finding all ratios from one given ratio

If sinθ = 3/5, find the values of all other trigonometric ratios of θ.

Since sinθ = opposite/hypotenuse = 3/5, take opposite = 3, hypotenuse = 5.
Adjacent = √(5² − 3²) = √(25−9) = √16 = 4.
cosθ = 4/5, tanθ = 3/4, cosecθ = 5/3, secθ = 5/4, cotθ = 4/3

Example 2 — Evaluating an expression using the angle table

Evaluate: 2tan²45° + cos²30° − sin²60°.

tan45° = 1, cos30° = √3/2, sin60° = √3/2.
= 2(1)² + (√3/2)² − (√3/2)² = 2 + 3/4 − 3/4 = 2

Example 3 — Proving an identity

Prove that (sinθ + cosθ)² + (sinθ − cosθ)² = 2.

Expand both squares: (sin²θ + 2sinθcosθ + cos²θ) + (sin²θ − 2sinθcosθ + cos²θ)
= 2sin²θ + 2cos²θ = 2(sin²θ + cos²θ) = 2(1) = 2 (using sin²θ+cos²θ=1) ✅

Example 4 — Using the complementary angle rule

Evaluate: sin63° / cos27°.

Since 63° + 27° = 90°, cos27° = cos(90°−63°) = sin63°.
So sin63°/cos27° = sin63°/sin63° = 1

Example 5 — Using a Pythagorean identity

If tanθ = 12/5, find the value of secθ.

Using 1 + tan²θ = sec²θ: sec²θ = 1 + (12/5)² = 1 + 144/25 = 169/25
secθ = √(169/25) = 13/5

Example 6 — Evaluating a mixed expression

Evaluate: (sin45° + cos45°) / (sec45° + cosec45°).

sin45° = cos45° = 1/√2. sec45° = cosec45° = √2.
Numerator = 1/√2 + 1/√2 = 2/√2 = √2. Denominator = √2 + √2 = 2√2.
Result = √2 / 2√2 = 1/2

6. Common Mistakes Students Make

⚠️ Mistake 1 — Mixing up opposite and adjacent. These labels depend on which angle you're measuring from — always re-check which side is opposite the angle in question before writing any ratio.

⚠️ Mistake 2 — Writing reciprocal ratios incorrectly. cosecθ = 1/sinθ, not cosθ — a very common mix-up caused by the similar-looking names.

⚠️ Mistake 3 — Misapplying the Pythagorean identities. Confusing 1+tan²θ=sec²θ with 1+cot²θ=cosec²θ, or trying to use sin²θ+cos²θ=1 where a tangent identity is actually needed.

⚠️ Mistake 4 — Using the complementary rule on angles that don't sum to 90°. sin(90°−θ)=cosθ only works for that exact complement — it cannot be applied to arbitrary angle pairs.

⚠️ Mistake 5 — Misremembering the standard angle table. Especially forgetting that tan90° and sec90° are undefined, or swapping the sin/cos columns for 30° and 60°.

⚠️ Mistake 6 — Cancelling trigonometric ratios like ordinary algebra. sinθ in a sum cannot be cancelled the same way as a common algebraic factor unless it's genuinely a multiplicative factor throughout — treat trig ratios as unified terms, not separable variables.

⚠️ Mistake 7 — Leaving surds unrationalised. 1/√3 and √3/3 are equal, but many mark schemes expect the rationalised form — always simplify at the final step.

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7. Quick Formula Recap Table

Concept	Formula
Basic ratios	sinθ=opp/hyp, cosθ=adj/hyp, tanθ=opp/adj
Reciprocal ratios	cosecθ=1/sinθ, secθ=1/cosθ, cotθ=1/tanθ
Quotient identities	tanθ=sinθ/cosθ, cotθ=cosθ/sinθ
Pythagorean identity 1	sin²θ + cos²θ = 1
Pythagorean identity 2	1 + tan²θ = sec²θ
Pythagorean identity 3	1 + cot²θ = cosec²θ
Complementary angles	sin(90°−θ)=cosθ, tan(90°−θ)=cotθ, sec(90°−θ)=cosecθ

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8. How to Revise This Chapter Effectively

📖 Derive the angle table instead of memorising it blindly. The 30°-60°-90° and 45°-45°-90° triangles generate every value in the table — knowing where the numbers come from makes them far harder to forget.

✍️ Say the three identities out loud, grouped by which ratio pair they connect. sin-cos, tan-sec, cot-cosec — this grouping mirrors how they're derived and makes recall much faster under time pressure.

🧮 Practise identity proofs by working from the more complex side toward the simpler one. Example 3 above shows this — expand, simplify, and land on the Pythagorean identity rather than guessing the path backward.

🎯 Double-check any complementary angle question by confirming the two angles genuinely sum to 90° before applying the shortcut — this single check prevents Mistake 4 above.

🔁 Revisit this page before your unit test. Bookmark it, and use the Quick Formula Recap Table (Section 7) as your final five-minute revision before the exam.

📚 Also Useful on MyTestSeries

🌐 Trusted External References

NCERT Official Website — download the official Class 10 Maths textbook
CBSE Academic — official syllabus & sample papers
Trigonometric Functions — Wikipedia — broader mathematical background
List of Trigonometric Identities — Wikipedia — identities beyond the Class 10 syllabus

9. Frequently Asked Questions

What are the six trigonometric ratios?

sinθ, cosθ, tanθ and their reciprocals cosecθ, secθ, cotθ — all defined as ratios between the three sides of a right triangle relative to angle θ.

What are the three Pythagorean trigonometric identities?

sin²θ+cos²θ=1, 1+tan²θ=sec²θ, and 1+cot²θ=cosec²θ — all derived from the Pythagoras theorem.

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

See the full table in Section 2 — sin goes from 0 to 1, cos goes from 1 to 0, and tan goes from 0 to undefined across these five angles.

What is the relationship between trigonometric ratios of complementary angles?

For angles adding to 90°: sin(90°−θ)=cosθ, cos(90°−θ)=sinθ, tan(90°−θ)=cotθ, and the corresponding reciprocal pairs.

What are the most common mistakes in Class 10 Trigonometry?

Mixing up opposite and adjacent sides, writing reciprocal ratios incorrectly, misapplying the Pythagorean identities, using the complementary rule on non-complementary angles, and misremembering the standard angle table.

Conclusion: One Triangle, Every Formula

Every formula in this chapter traces back to the same right triangle and the same angle θ. The six ratios describe its sides, the three identities describe how those ratios relate to each other through Pythagoras, and the complementary angle rules describe what happens when you look at the triangle from its other acute angle. Once that single picture is solid, the rest of the chapter is substitution and simplification.

Rework the six solved examples above without checking the answers first, derive the standard angle table from the 30-60-90 and 45-45-90 triangles rather than memorising it cold, and always rationalise your final surd. That combination is what turns a page of formulas into consistent exam marks.

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Tags: Introduction to Trigonometry Class 10, Trigonometric Ratios, Trigonometric Identities, Standard Angle Table, Complementary Angles, CBSE 2026, NCERT Class 10 Maths Chapter 8

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