Class 10 / Maths
Class 10 Maths Introduction to Trigonometry Notes
Notes on trigonometric ratios, standard angles, identities, and right-triangle applications.
Topic introduction
Introduction to Trigonometry connects right triangles with ratios. The key is to choose the angle first, label perpendicular, base, and hypotenuse correctly, and then use identities or standard values as needed.
Key points
- Trigonometric ratios are defined for acute angles of a right triangle.
- sin theta = perpendicular/hypotenuse, cos theta = base/hypotenuse, and tan theta = perpendicular/base.
- The reciprocal ratios are cosec theta, sec theta, and cot theta.
- Important identities include sin^2 theta + cos^2 theta = 1, 1 + tan^2 theta = sec^2 theta, and 1 + cot^2 theta = cosec^2 theta.
Definitions
Hypotenuse: The longest side of a right triangle, opposite the right angle.
Perpendicular: The side opposite the chosen acute angle.
Base: The side adjacent to the chosen acute angle, other than the hypotenuse.
Trigonometric identity: An equation involving trigonometric ratios that is true for all valid values of the angle.
Formulas
- sin A = perpendicular/hypotenuse
- cos A = base/hypotenuse
- tan A = perpendicular/base
- sin^2 A + cos^2 A = 1
- 1 + tan^2 A = sec^2 A
- 1 + cot^2 A = cosec^2 A
Examples
- If perpendicular = 6 and hypotenuse = 10, then sin A = 6/10 = 3/5.
- If tan A = 3/4, use a 3-4-5 triangle to get sin A = 3/5 and cos A = 4/5.
- sec^2 A - tan^2 A = 1 from the identity 1 + tan^2 A = sec^2 A.
Ratios in a right triangle
Choose the angle first. The side opposite the chosen angle is the perpendicular, the side adjacent to it is the base, and the longest side is the hypotenuse.
Most mistakes happen when students change the angle but keep the same perpendicular and base. Always redraw or relabel if needed.
Standard angle values
Memorize values for 0, 30, 45, 60, and 90 degrees. These values are used repeatedly in direct questions and simplification problems.
Use identities to simplify expressions before substituting values whenever possible.
Common mistakes
- Changing the angle but not relabelling perpendicular and base.
- Confusing reciprocal ratios such as sin and cosec.
- Writing tan A as base/perpendicular instead of perpendicular/base.
- Using identities without checking which expression needs simplification.
- Forgetting that standard values are for specific angles such as 30, 45, 60, and 90 degrees.
Quick practice
- If sin A = 3/5, find cos A and tan A.
- Evaluate 2 sin 30 degrees + cos 60 degrees.
- Prove that 1 + tan^2 A = sec^2 A.
- Simplify sin^2 A + cos^2 A + tan^2 A.
Related practice
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