Trigonometry Cheat Sheet: Essential Formulas and Identities, Study Guides, Projects, Research of Mathematics

Download Trigonometry Cheat Sheet: Essential Formulas and Identities and more Study Guides, Projects, Research Mathematics in PDF only on Docsity! Trigonometry Cheat Sheet: Essential Formulas and Identities Trigonometry is an important branch of mathematics that deals with the relationship between the sides and angles of triangles. It has a wide range of applications in various fields such as engineering, physics, and navigation. In this tutorial, we will provide a comprehensive overview of the essential formulas and identities in trigonometry. Basic Concepts Before delving into the formulas and identities, it is important to understand some basic concepts in trigonometry. Right Triangle A right triangle is a triangle with one angle measuring 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. Trigonometric Functions Trigonometric functions are ratios of the sides of a right triangle. The three basic trigonometric functions are sine, cosine, and tangent, and they are defined as follows: Sine (sin): ratio of the opposite side to the hypotenuse Cosine (cos): ratio of the adjacent side to the hypotenuse Tangent (tan): ratio of the opposite side to the adjacent side Unit Circle The unit circle is a circle with a radius of 1 unit, centered at the origin on a coordinate plane. It is used to define the values of trigonometric functions for any angle. Essential Formulas and Identities Now, let's dive into the essential formulas and identities in trigonometry. Pythagorean Theorem The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). This can also be written in the form of an equation: c² = a² + b² Trigonometric Identities Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. These identities are useful in simplifying trigonometric expressions and solving equations. Here are some of the most commonly used identities: Pythagorean Identities: sin²θ + cos²θ = 1 1 + tan²θ = sec²θ cot²θ + 1 = csc²θ Sum and Difference Identities: sin(α ± β) = sinαcosβ ± cosαsinβ cos(α ± β) = cosαcosβ ∓ sinαsinβ tan(α ± β) = (tanα ± tanβ) / (1 ∓ tanαtanβ) Double Angle Identities: sin2θ = 2sinθcosθ cos2θ = cos²θ - sin²θ tan2θ = 2tanθ / (1 - tan²θ) Half Angle Identities: sin(θ/2) = ±√[(1 - cosθ) / 2] cos(θ/2) = ±√[(1 + cosθ) / 2] tan(θ/2) = ±√[(1 - cosθ) / (1 + cosθ)] Product-to-Sum Identities: sinαsinβ = (1/2)[cos(α - β) - cos(α + β)] cosαcosβ = (1/2)[cos(α - β) + cos(α + β)] sinαcosβ = (1/2)[sin(α + β) + sin(α - β)] Sum-to-Product Identities: sinα + sinβ = 2sin[(α + β)/2]cos[(α - β)/2] sinα - sinβ = 2cos[(α + β)/2]sin[(α - β)/2] cosα + cosβ = 2cos[(α + β)/2]cos[(α - β)/2] cosα - cosβ = -2sin[(α + β)/2]sin[(α - β)/2] Law of Sines and Cosines The law of sines and cosines are two important formulas used in solving triangles.