Trigonometry- study Guide notes, Summaries of Mathematics

Download Trigonometry- study Guide notes and more Summaries Mathematics in PDF only on Docsity! Trigonometry Introduction Trigonometry is a branch of mathematics that deals with the study of angles and their relationships with sides of triangles. It has wide applications in fields such as engineering, physics, astronomy, and many others. Topics in Trigonometry: Basic concepts of trigonometry Trigonometric functions Trigonometric identities Inverse trigonometric functions Applications of trigonometry Solving trigonometric equations Basic Concepts of Trigonometry: a. Angles: An angle is formed when two lines or rays meet at a point. The measure of an angle is usually expressed in degrees or radians. A degree is 1/360th of a circle, and a radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. b. Right triangles: A right triangle is a triangle in which one of the angles is a right angle (90 degrees). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. Page 1 of 1 c. Trigonometric ratios: Trigonometric ratios are ratios of the lengths of the sides of a right triangle. The three basic trigonometric ratios are sine, cosine, and tangent. They are defined as follows: Sine: sin(theta) = opposite/hypotenuse Cosine: cos(theta) = adjacent/hypotenuse Tangent: tan(theta) = opposite/adjacent d. Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In other words, a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the legs. Trigonometric Functions: a. Sine function: The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is denoted by sin(theta). The sine function is periodic with a period of 2π, which means that it repeats itself every 2π units. b. Cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle. It is denoted by cos(theta). The cosine function is also periodic with a period of 2π. c. Tangent function: The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. It is denoted by tan(theta). The tangent function is not periodic. d. Cosecant function: The cosecant function is the reciprocal of the sine function. It is denoted by csc(theta). The cosecant function is not periodic. e. Secant function: The secant function is the reciprocal of the cosine function. It is denoted by sec(theta). The secant function is not periodic. f. Cotangent function: The cotangent function is the reciprocal of the tangent function. It is denoted by cot(theta). The cotangent function is periodic with a period of π. Page 1 of 2 Solving Trigonometric Equations: a. Linear equations: Linear trigonometric equations can be solved using algebraic methods such as substitution and simplification. b. Quadratic equations: Quadratic trigonometric equations can be solved by factoring, completing the square, or using the quadratic formula. c. Other types of equations : Other types of trigonometric equations, such as equations involving multiple angles or equations involving inverse trigonometric functions, can be solved using various techniques such as the sum and difference formulas, the double angle formula, or the use of trigonometric identities. Trigonometric Functions and Graphs: a. The unit circle: The unit circle is a circle with a radius of one unit centered at the origin of a coordinate plane. It is used to define the values of the trigonometric functions for all angles. b. Graphing trigonometric functions: The graphs of the six trigonometric functions can be generated using a unit circle or by plotting points on a coordinate plane. The graphs of these functions exhibit periodic behavior and have certain properties such as amplitude, period, and phase shift. c. Transformations of trigonometric functions: Trigonometric functions can be transformed by shifting, stretching, or reflecting their graphs. These transformations affect the amplitude, period, and phase shift of the functions. Trigonometric Identities and Equations: a. Proving trigonometric identities: Trigonometric identities can be proven using various techniques such as algebraic manipulation, factoring, and the use of trigonometric identities. b. Solving trigonometric equations: Trigonometric equations can be solved using algebraic methods, the use of trigonometric identities, or by graphing the functions involved. Page 1 of 5 Applications of Trigonometry in Real Life: Trigonometry has numerous applications in real life, some of which are listed below: a. Architecture and construction: Trigonometry is used in architecture and construction to determine angles, distances, and heights of structures. b. Art and design: Trigonometry is used in art and design to create various shapes and patterns. c. Sports: Trigonometry is used in sports such as football and basketball to calculate the trajectory of a ball. d. Music: Trigonometry is used in music to create sound waves and to study the frequency and amplitude of sound. e. Medical imaging: Trigonometry is used in medical imaging such as MRI and CT scans to generate 3D images of the body. f. Video games and animation: Trigonometry is used in video games and animation to create realistic movements and special effects. Conclusion : Trigonometry is an important branch of mathematics that deals with the study of triangles and their properties. It has numerous applications in various fields such as navigation, astronomy, engineering, physics, and many others. The study of trigonometry involves learning various concepts such as the trigonometric functions, identities, equations, and their applications. It also involves the use of mathematical tools such as the unit circle, graphing calculators, and various trigonometric formulas and techniques. Page 1 of 6