Master the angles, ratios, and identities that form the foundation of advanced mathematics and real-world problem solving.
Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. The word comes from Greek words "trigonon" (triangle) and "metron" (measure).
In Class 10 CBSE, we focus on right-angled triangles and explore six fundamental ratios that help us solve problems in fields like physics, engineering, architecture, and astronomy.
Consider a right-angled triangle with one acute angle θ (theta). We define three sides relative to this angle:
Sine represents the ratio of the side opposite to the angle and the hypotenuse.
Cosine represents the ratio of the side adjacent to the angle and the hypotenuse.
Tangent is the ratio of the opposite side to the adjacent side. Also: tan θ = sin θ / cos θ
Cotangent is the reciprocal of tangent. Also: cot θ = 1 / tan θ = cos θ / sin θ
Secant is the reciprocal of cosine: sec θ = 1 / cos θ
Cosecant is the reciprocal of sine: cosec θ = 1 / sin θ
These are the most important values you'll use in problem-solving. Memorize them—they appear everywhere in Class 10 exams and beyond!
| Angle (θ) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Not Defined |
| cot θ | Not Defined | √3 | 1 | 1/√3 | 0 |
| sec θ | 1 | 2/√3 | √2 | 2 | Not Defined |
| cosec θ | Not Defined | 2 | √2 | 2/√3 | 1 |
| Value | Exact | Approximate |
|---|---|---|
| 1/√2 | 1/√2 | ≈ 0.707 |
| √3/2 | √3/2 | ≈ 0.866 |
| 1/√3 | 1/√3 | ≈ 0.577 |
| √3 | √3 | ≈ 1.732 |
| √2 | √2 | ≈ 1.414 |
Identities are equations that are true for all values of the variable. These are essential for simplifying expressions and solving complex problems.
This is the fundamental identity derived from the Pythagorean theorem. From this, we get two more:
For complementary angles (angles that sum to 90°):
Calculate the height of buildings, towers, and mountains without physically measuring them. Uses angle of elevation and angle of depression.
Example: Finding the height of a tower by measuring the angle of elevation from a known distance.
Ships and aircraft use trigonometry to calculate distances, bearings, and routes across the globe.
Example: Determining the shortest path between two points on Earth's surface.
Designing structures, calculating load distributions, and determining angles for optimal stability.
Example: Calculating roof slopes, bridge angles, and structural stress points.
Measuring distances to stars, calculating planetary orbits, and predicting celestial events.
Example: Using parallax method to find distances to nearby stars.
Analyzing periodic motion, sound waves, light waves, and electromagnetic radiation.
Example: Describing simple harmonic motion and AC circuits.
Rotating objects, calculating camera angles, and rendering 3D scenes in games and simulations.
Example: Game engines use trig for character movement and camera positioning.
| Category | Formula |
|---|---|
| Basic Ratios | sin θ = P/H, cos θ = B/H, tan θ = P/B |
| Reciprocals | cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ |
| Pythagorean | sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ |
| Complementary | sin(90° - θ) = cos θ, cos(90° - θ) = sin θ |
| Quotient | tan θ = sin θ/cos θ, cot θ = cos θ/sin θ |