📚 Class 10 Mathematics – Chapter: Introduction to Trigonometry
Quick Revision Sheet
1️⃣ Basic Definitions
| Term | Meaning |
|---|
| Trigonometry | Study of relationships between the sides and angles of a triangle (mainly right‑angled triangles). |
| Angle of Elevation / Depression | Angle measured upward / downward from the horizontal line of sight. |
| Reference Angle | The acute angle formed by the terminal side of any angle and the x‑axis. |
| Quadrant | One of the four parts of the coordinate plane (I, II, III, IV). |
| Trigonometric Ratio | Ratio of two sides of a right‑angled triangle for a given acute angle. |
2️⃣ Trigonometric Ratios (for an acute angle θ)
| Ratio | Symbol | Formula (using right‑angled triangle) |
|---|
| Sine | sin θ | opposite / hypotenuse |
| Cosine | cos θ | adjacent / hypotenuse |
| Tangent | tan θ | opposite / adjacent |
| Cosecant | cosec θ | 1 / sin θ = hypotenuse / opposite |
| Secant | sec θ | 1 / cos θ = hypotenuse / adjacent |
| Cotangent | cot θ | 1 / tan θ = adjacent / opposite |
Mnemonic – “SOH‑CAH‑TOA”
sin θ = Opposite/Hypotenuse cos θ = Adjacent/Hypotenuse tan θ = Opposite/Adjacent
3️⃣ Fundamental Identities
| Identity | Expression |
|---|
| Pythagorean | sin²θ + cos²θ = 1 |
| Derived Pythagorean | tan²θ + 1 = sec²θ 1 + cot²θ = cosec²θ |
| Reciprocal | cosec θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ |
| Quotient | tan θ = sin θ / cos θ cot θ = cos θ / sin θ |
| Co‑function (Complementary Angles) | sin(90°‑θ) = cos θ cos(90°‑θ) = sin θ<br>tan(90°‑θ) = cot θ cot(90°‑θ) = tan θ<br>sec(90°‑θ) = cosec θ cosec(90°‑θ) = sec θ |
4️⃣ Angle‑Addition & Subtraction Formulas
| Formula | Expression |
|---|
| Sine | sin(A ± B) = sinA·cosB ± cosA·sinB |
| Cosine | cos(A ± B) = cosA·cosB ∓ sinA·sinB |
| Tangent | tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA·tanB) |
(Upper sign for “+”, lower sign for “–”)
5️⃣ Double‑Angle Formulas
| Formula | Expression |
|---|
| Sine | sin 2θ = 2 sinθ cosθ |
| Cosine | cos 2θ = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ |
| Tangent | tan 2θ = 2 tanθ / (1 – tan²θ) |
6️⃣ Half‑Angle Formulas
| Formula | Expression |
|---|
| Sine | sin(θ/2) = ±√[(1 – cosθ)/2] |
| Cosine | cos(θ/2) = ±√[(1 + cosθ)/2] |
| Tangent | tan(θ/2) = ±√[(1 – cosθ)/(1 + cosθ)] = (1 – cosθ)/sinθ = sinθ/(1 + cosθ) |
Sign (±) depends on the quadrant of θ/2.
7️⃣ Values of Trigonometric Ratios for Common Angles
| θ (degrees) | sin θ | cos θ | tan θ | cosec θ | sec θ | cot θ |
|---|
| 0° | 0 | 1 | 0 | – | 1 | – |
| 30° | ½ | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | √3/2 | ½ | √3 | 2/√3 | 2 | 1/√3 |
| 90° | 1 | 0 | – | 1 | – | 0 |
(“–” indicates the ratio is undefined.)
8️⃣ Solving Triangles – Quick Steps
- Identify the given sides/angles.
- Choose the appropriate trigonometric ratio (sin, cos, tan).
- Write the equation using the ratio.
- Solve for the unknown side or angle.
- Use complementary‑angle or reciprocal identities if the required ratio is not directly given.
9️⃣ Important Tips for Exams
| Tip | Explanation |
|---|
| Know the sign of each ratio in all quadrants | Q I (+,+,+), Q II (+,‑,+), Q III (‑,‑,+), Q IV (+,‑,‑). |
| Memorise the 5 special angles (0°,30°,45°,60°,90°) | Saves time when evaluating expressions. |
| Convert degrees ↔ radians when required | π rad = 180°. |
| Check for “undefined” | tan θ undefined when cos θ = 0 (θ = 90°, 270° …). |
| Use the “±” rule wisely in half‑angle formulas | Determine the sign from the quadrant of the resulting angle. |
📌 Summary
- Trigonometric ratios relate sides of a right‑angled triangle to its acute angles.
- Core identities (Pythagorean, reciprocal, quotient, co‑function) are the foundation for all manipulations.
- Angle‑addition, double‑angle, and half‑angle formulas help simplify complex expressions and solve equations.
- Remember the standard values for 0°, 30°, 45°, 60°, 90° and the sign of each ratio in the four quadrants.
Keep this sheet handy while revising – it contains every formula you’ll need for Class 10 Introduction to Trigonometry! 🚀