Explore the basics of trigonometry, including angles, triangles, and the fundamental trigonometric ratios: sine, cosine, and tangent.
Introduction to Trigonometry - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Introduction to Trigonometry aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Define Trigonometry.
Trigonometry is the study of relationships between the sides and angles of a triangle. It's derived from Greek words 'tri' (three), 'gon' (sides), and 'metron' (measure).
Six trigonometric ratios.
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). Each is a ratio of sides in a right-angled triangle.
sin A = opposite/hypotenuse.
Sine of angle A is the ratio of the length of the side opposite to angle A to the hypotenuse. Example: In a right triangle with opposite side 3 and hypotenuse 5, sin A = 3/5.
cos A = adjacent/hypotenuse.
Cosine of angle A is the ratio of the length of the adjacent side to angle A to the hypotenuse. Example: Adjacent side 4, hypotenuse 5, cos A = 4/5.
tan A = opposite/adjacent.
Tangent of angle A is the ratio of the opposite side to the adjacent side. It's also sin A/cos A. Example: Opposite 3, adjacent 4, tan A = 3/4.
Pythagorean identity: sin²A + cos²A = 1.
This identity shows the relationship between sin and cos of an angle. It's derived from the Pythagorean theorem applied to a right-angled triangle.
1 + tan²A = sec²A.
This identity relates tangent and secant of an angle. It's useful for converting between trigonometric functions in equations.
1 + cot²A = cosec²A.
This identity relates cotangent and cosecant of an angle, another way to express relationships between trigonometric functions.
Trigonometric ratios of 0°, 30°, 45°, 60°, 90°.
Memorize the values: sin 0°=0, sin 30°=1/2, sin 45°=1/√2, sin 60°=√3/2, sin 90°=1. Similar patterns exist for cos, tan, etc.
Real-world applications of trigonometry.
Trigonometry is used in architecture, astronomy, engineering, and physics to calculate distances and angles in real-world scenarios.
Reciprocal relationships.
cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A. These show how the functions are interconnected.
tan A = sin A / cos A.
This formula allows conversion between tan, sin, and cos, useful for simplifying trigonometric expressions.
cot A = cos A / sin A.
Similar to tan A, this formula relates cotangent to sine and cosine, aiding in expression simplification.
Trigonometric identities simplification.
Using identities to simplify expressions is a key skill. Practice converting all terms to sin and cos to simplify.
Angle of elevation and depression.
These concepts use trigonometry to find heights and distances, common in real-life problems and exams.
Misconception: sin⁻¹A = 1/sin A.
sin⁻¹A is the inverse sine function, not the reciprocal. The reciprocal is cosec A.
Memory hack for ratios.
Use 'SOHCAHTOA' to remember: Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent.
Value range of sin and cos.
sin A and cos A values range between -1 and 1, but for acute angles (0° to 90°), they're between 0 and 1.
sec and cosec range.
For acute angles, sec A ≥ 1 and cosec A ≥ 1, since they're reciprocals of cos and sin, which are ≤1.
Solving right-angled triangles.
Given one side and one acute angle, or two sides, you can find all other sides and angles using trigonometric ratios.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
