In short: The complete Class 10 trigonometry cheat sheet. Standard value table for all angles, memory tricks to recall values instantly, every identity you need, complementary angle rules, and Heights & Distances formulas with diagrams. Bookmark this page — you will use it before every maths exam.
Standard Trigonometric Values Table (0° to 90°)
| Angle | sin | cos | tan | cosec | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | Not defined | 1 | Not defined |
| 30° | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | 1/√2 | 1/√2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | 1 | 0 | Not defined | 1 | Not defined | 0 |
Memory Trick — The Finger Method
| Finger (from thumb) | Angle | sin value = √(finger number)/2 |
|---|---|---|
| Thumb (0) | 0° | √0/2 = 0 |
| Index (1) | 30° | √1/2 = 1/2 |
| Middle (2) | 45° | √2/2 = 1/√2 |
| Ring (3) | 60° | √3/2 |
| Little (4) | 90° | √4/2 = 1 |
For cos: Reverse the order — cos 0° = √4/2 = 1, cos 30° = √3/2, etc.
Quick Trick Table
| To get | From sin table |
|---|---|
| cos θ | = sin (90° - θ), i.e., read sin table in reverse |
| tan θ | = sin θ / cos θ |
| cosec θ | = 1 / sin θ |
| sec θ | = 1 / cos θ |
| cot θ | = 1 / tan θ = cos θ / sin θ |
Decimal Values (For Quick Checking)
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0.000 | 1.000 | 0.000 |
| 30° | 0.500 | 0.866 | 0.577 |
| 45° | 0.707 | 0.707 | 1.000 |
| 60° | 0.866 | 0.500 | 1.732 |
| 90° | 1.000 | 0.000 | ∞ |
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All Trigonometric Identities (Class 10)
Fundamental Identities
| # | Identity | Derived Forms |
|---|---|---|
| 1 | sin²θ + cos²θ = 1 | sin²θ = 1 - cos²θ; cos²θ = 1 - sin²θ |
| 2 | 1 + tan²θ = sec²θ | tan²θ = sec²θ - 1; sec²θ - tan²θ = 1 |
| 3 | 1 + cot²θ = cosec²θ | cot²θ = cosec²θ - 1; cosec²θ - cot²θ = 1 |
Reciprocal Relationships
| Pair | Relationship |
|---|---|
| sin θ and cosec θ | sin θ = 1/cosec θ; cosec θ = 1/sin θ |
| cos θ and sec θ | cos θ = 1/sec θ; sec θ = 1/cos θ |
| tan θ and cot θ | tan θ = 1/cot θ; cot θ = 1/tan θ |
Quotient Relationships
| Formula | Expression |
|---|---|
| tan θ | sin θ / cos θ |
| cot θ | cos θ / sin θ |
Complementary Angles (90° - θ)
| Original | Complementary Form | Rule |
|---|---|---|
| sin(90° - θ) | cos θ | sin ↔ cos |
| cos(90° - θ) | sin θ | cos ↔ sin |
| tan(90° - θ) | cot θ | tan ↔ cot |
| cot(90° - θ) | tan θ | cot ↔ tan |
| sec(90° - θ) | cosec θ | sec ↔ cosec |
| cosec(90° - θ) | sec θ | cosec ↔ sec |
Memory trick: Co-function pairs swap. sin ↔ cos, tan ↔ cot, sec ↔ cosec.
Heights and Distances — Key Terms
| Term | Meaning | Diagram Position |
|---|---|---|
| Angle of Elevation | Angle above horizontal line of sight | Observer looks UP |
| Angle of Depression | Angle below horizontal line of sight | Observer looks DOWN |
| Line of Sight | Straight line from eye to object | Hypotenuse of triangle |
| Horizontal | Level line from observer | Base or reference |
Heights and Distances — Formula Table
| Scenario | Given | Formula | Find |
|---|---|---|---|
| Height of object | Angle of elevation (θ), distance from base (d) | h = d × tan θ | Height (h) |
| Distance from base | Angle of elevation (θ), height (h) | d = h / tan θ | Distance (d) |
| Height of tower (two angles) | Angles θ₁ and θ₂ from points at distance d apart | h = d × tan θ₁ × tan θ₂ / (tan θ₁ - tan θ₂) | Height (h) |
| Width of river | Angle of depression (θ), height of cliff (h) | w = h / tan θ | Width (w) |
| Distance between two objects | Angle of depression θ₁ and θ₂ from top, height h | Distance = h(cot θ₂ - cot θ₁) | Gap between objects |
Common Height & Distance Setups
| Setup | Triangle Type | Key Ratio |
|---|---|---|
| Person looking at top of tower | Right triangle (ground level) | tan θ = height/distance |
| Person on top looking down | Right triangle (from top) | tan θ = height/distance (angle of depression = angle of elevation) |
| Two positions, same object | Two right triangles sharing vertical side | Use both tan values |
| Shadow problems | Right triangle (sun angle) | tan θ = height of object/shadow length |
| Lighthouse/cliff problems | Right triangle (vertical height) | tan θ = cliff height/distance to ship |
Identity Proof — Common Patterns
| What to prove | Strategy |
|---|---|
| LHS = RHS (both sides given) | Simplify the more complex side |
| Expression = constant | Convert everything to sin and cos |
| Expression involves sec, cosec | Use reciprocal forms: 1/cos, 1/sin |
| Expression has tan²θ or cot²θ | Use sec²θ - 1 or cosec²θ - 1 |
| Numerator/Denominator form | Take LCM, simplify, factorize |
| (a + b)(a - b) pattern | Use a² - b² identity |
| LHS has addition/subtraction of fractions | Combine into single fraction |
Useful Algebraic Identities in Trig Proofs
| Identity | Application in Trig |
|---|---|
| a² - b² = (a+b)(a-b) | Factoring sin²θ - cos²θ type expressions |
| (a+b)² = a² + 2ab + b² | Expanding (sinθ + cosθ)² |
| (a-b)² = a² - 2ab + b² | Expanding (secθ - tanθ)² |
| a³ + b³ = (a+b)(a²-ab+b²) | Factoring sin³θ + cos³θ |
| a³ - b³ = (a-b)(a²+ab+b²) | Factoring sec³θ - tan³θ |
Value Verification Shortcut
| If you calculate and get... | Check against |
|---|---|
| sin θ > 1 | Impossible. Recalculate. |
| cos θ > 1 | Impossible. Recalculate. |
| sin θ < 0 (for 0° < θ < 90°) | Impossible. Recalculate. |
| tan 45° ≠ 1 | Error somewhere |
| sin²θ + cos²θ ≠ 1 | Your values are wrong |
Trigonometric Ratios in Right Triangle
| Ratio | Formula (in right triangle) |
|---|---|
| sin θ | Opposite / Hypotenuse = P/H |
| cos θ | Adjacent / Hypotenuse = B/H |
| tan θ | Opposite / Adjacent = P/B |
| cosec θ | Hypotenuse / Opposite = H/P |
| sec θ | Hypotenuse / Adjacent = H/B |
| cot θ | Adjacent / Opposite = B/P |
Memory trick: SOH-CAH-TOA → Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Another popular mnemonic: Pandit Badri Prasad, Har Har Bole → P/H, B/H, P/B for sin, cos, tan.
Frequently Asked Questions
Q1. How do I memorize the sin-cos-tan table? Use the finger trick: for sin, assign 0-4 to your five fingers. sin 0° = √0/2, sin 30° = √1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = √4/2. For cos, reverse the order. Tan = sin/cos.
Q2. How many trigonometry identities are in Class 10? There are 3 fundamental identities (sin² + cos² = 1, 1 + tan² = sec², 1 + cot² = cosec²), 6 reciprocal/quotient relations, and 6 complementary angle rules. Total: about 15 core formulas.
Q3. What is the difference between angle of elevation and angle of depression? Angle of elevation: you look UP from horizontal. Angle of depression: you look DOWN from horizontal. Key fact: angle of elevation from A to B equals angle of depression from B to A (alternate interior angles).
Q4. How many marks does trigonometry carry in Class 10 boards? Trigonometry (Introduction + Applications) carries about 12-14 marks in CBSE Class 10 Maths. This typically includes 1 proof question (3-4 marks), 1 Heights & Distances problem (4-5 marks), and 2-3 MCQs.
Q5. Do I need to memorize decimal values of sin, cos, tan? Not for board exams — answers are accepted in fraction/surd form (like √3/2 or 1/√2). But knowing approximate decimals helps verify calculator-based or MCQ answers quickly.
Q6. Which identity should I start with when proving a trig equation? Start by converting everything to sin and cos. Then look for patterns: can you use sin² + cos² = 1? Can you factor? Can you take LCM? Most Class 10 proofs need just 3-4 steps if you pick the right starting identity.
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