All trigonometric identities

Students are taught about trig identities or trigonometric identities in school and are an important part of higher-level mathematics. So to help you understand and learn all trig identities we have explained here all the concepts of trigonometry. As a student, you would find the trig identity sheet we have provided here useful. So you can download and print the identities PDF and use it anytime to solve the equations. Trigonometry is an important branch of mathematics that deals with relationships between the lengths and angles of triangles. It is quite an old concept and was first used in the 3rd century BC. This branch of mathematics is related to planar right-triangles (or the right-triangles in a two-dimensional plane with one angle equal to 90 degrees). There are some other branches where trigonometry has contributed immensely in its growth and development. Some of its fields of application are ; • In music: It can be used to develop music digitally, through computer music. • In aviation: it is of vital importance to lead an aircraft in the right direction. For instance, if the wind speed and the angle of the aircraft are known, it can be used to determine the direction of the aircraft. • In criminology – trigonometry can also be used in criminology where it is used to calculate various important determinants of a crime scene, such as the trajectory of a projectile, how an object falls, etc. • Mathematics: Trigonometry is one of the most important branches of mathema...

\( \newcommand\) • • • • • Focus Questions The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts mo- tivated by these questions and be able to write precise, coherent answers to these questions. • What is an identity? • How do we verify an identity? Consider the trigonometric equation \(\sin(2x) = \cos(x)\). Based on our current knowledge, an equation like this can be difficult to solve exactly because the periods of the functions involved are different. What will allow us to solve this equation relatively easily is a trigonometric identity, and we will explicitly solve this equation in a subsequent section. This section is an introduction to trigonometric identities. As we discussed in Section 2.6, a mathematical equation like \(x^ = 1\) since this is true for all real number values of \(x\). So while we solve equations to determine when the equality is valid, there is no reason to solve an identity since the equality in an identity is always valid. Every identity is an equation, but not every equation is an identity. To know that an equation is an identity it is necessary to provide a convincing argument that the two expressions in the equation are always equal to each other. Such a convincing argument is called a proof and we use proofs to verify trigonometric identities. Beginning Activity • Use a graphing utility to draw the graph of \(y = \cos(x - \dfrac)\) and \(\sin(x)\) ...

\( \newcommand\) • • • • • • • • • • • Radian Measure To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, \(radians\) are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. Given an angle \(θ\), let \(s\) be the length of the corresponding arc on the unit circle (Figure). We say the angle corresponding to the arc of length 1 has radian measure 1. Figure \(\PageIndex\): The radian measure of an angle \(θ\) is the arc length \(s\) of the associated arc on the unit circle. Since an angle of \(360°\) corresponds to the circumference of a circle, or an arc of length \(2π\), we conclude that an angle with a degree measure of \(360°\) has a radian measure of \(2π\). Similarly, we see that \(180°\) is equivalent to \(\pi\) radians. Table shows the relationship between common degree and radian values. The Six Basic Trigonometric Functions Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship among the sides and angles of a triangle. To define the trigonometric functions, first consider the unit circle centered at the origin and a point \(P=(x,y)\) on the unit circle. Let \(θ\) be an angle with an initial side that lies al...

\( \newcommand\) • • • • • • • • Learning Objectives • Use the fundamental identities to solve trigonometric equations. • Express trigonometric expressions in simplest form. • Solve trigonometric equations by factoring. • Solve trigonometric equations by using the Quadratic Formula. Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. The legend is that he calculated the height of the similar triangles, which he developed by measuring the shadow of his staff. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle of elevation and the angle of depression are found using similar triangles. Figure \(\PageIndex\): Egyptian pyramids standing near a modern city. (credit: Oisin Mulvihill) In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids. Solving Linear Trigonometric Equations in Sine and Cosine Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just...

Trigonometric Identities Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Trigonometric Identities are true for every value of variables occurring on both sides of an equation. Geometrically, these identities involve certain trigonometric functions (such as Sine, cosine and tangent are the primary trigonometry functions whereas cotangent, secant and cosecant are the other three functions. The trigonometric identities are based on all the six trig functions. Check Table of Contents: • • • • • • • • • • • • • • • • • • What are Trigonometric Identities? Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation. There are various distinct trigonometric identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle . All the trigonometric identities are based on the six trigonometric ratios. They are sine, cosine, tangent, cosecant, secant, and cotangent. All these trigonometric ratios are defined using the sides of the right triangle, such as an adjacent side, opposite side, and hypotenuse side. All the fundamental trigonometric identities are derived from the six trigonometric ratios. Trigonometric Identities PDF Click here to download the PDF of trigonometry identities of all functions such as sin, cos, tan and so on. Download PDF List o...

\( \newcommand\) • • • • • • • Pythagorean, Tangent, and Reciprocal Identities used to find values of functions. Trigonometric Identities You can use the Pythagorean, Tangent and Let's solve the following problems using trigonometric identities. • Given that \(\cos\theta =\dfrac\) Example \(\PageIndex\) Find the values of the other five trigonometric functions. Example \(\PageIndex\) from the Review • In which quadrants is the sine value positive? Negative? • In which quadrants is the cosine value positive? Negative? • In which quadrants is the tangent value positive? Negative? Find the values of the other five trigonometric functions of \(\theta \). • \(\sin\theta =\dfrac\) quadrant, what is \(\sec(−\theta )\)?

\( \newcommand\) • • • • • • • Pythagorean, Tangent, and Reciprocal Identities used to find values of functions. Trigonometric Identities You can use the Pythagorean, Tangent and Let's solve the following problems using trigonometric identities. • Given that \(\cos\theta =\dfrac\) Example \(\PageIndex\) Find the values of the other five trigonometric functions. Example \(\PageIndex\) from the Review • In which quadrants is the sine value positive? Negative? • In which quadrants is the cosine value positive? Negative? • In which quadrants is the tangent value positive? Negative? Find the values of the other five trigonometric functions of \(\theta \). • \(\sin\theta =\dfrac\) quadrant, what is \(\sec(−\theta )\)?

Students are taught about trig identities or trigonometric identities in school and are an important part of higher-level mathematics. So to help you understand and learn all trig identities we have explained here all the concepts of trigonometry. As a student, you would find the trig identity sheet we have provided here useful. So you can download and print the identities PDF and use it anytime to solve the equations. Trigonometry is an important branch of mathematics that deals with relationships between the lengths and angles of triangles. It is quite an old concept and was first used in the 3rd century BC. This branch of mathematics is related to planar right-triangles (or the right-triangles in a two-dimensional plane with one angle equal to 90 degrees). There are some other branches where trigonometry has contributed immensely in its growth and development. Some of its fields of application are ; • In music: It can be used to develop music digitally, through computer music. • In aviation: it is of vital importance to lead an aircraft in the right direction. For instance, if the wind speed and the angle of the aircraft are known, it can be used to determine the direction of the aircraft. • In criminology – trigonometry can also be used in criminology where it is used to calculate various important determinants of a crime scene, such as the trajectory of a projectile, how an object falls, etc. • Mathematics: Trigonometry is one of the most important branches of mathema...

\( \newcommand\) • • • • • Focus Questions The following questions are meant to guide our study of the material in this section. After studying this section, we should understand the concepts mo- tivated by these questions and be able to write precise, coherent answers to these questions. • What is an identity? • How do we verify an identity? Consider the trigonometric equation \(\sin(2x) = \cos(x)\). Based on our current knowledge, an equation like this can be difficult to solve exactly because the periods of the functions involved are different. What will allow us to solve this equation relatively easily is a trigonometric identity, and we will explicitly solve this equation in a subsequent section. This section is an introduction to trigonometric identities. As we discussed in Section 2.6, a mathematical equation like \(x^ = 1\) since this is true for all real number values of \(x\). So while we solve equations to determine when the equality is valid, there is no reason to solve an identity since the equality in an identity is always valid. Every identity is an equation, but not every equation is an identity. To know that an equation is an identity it is necessary to provide a convincing argument that the two expressions in the equation are always equal to each other. Such a convincing argument is called a proof and we use proofs to verify trigonometric identities. Beginning Activity • Use a graphing utility to draw the graph of \(y = \cos(x - \dfrac)\) and \(\sin(x)\) ...

\( \newcommand\) • • • • • • • • Learning Objectives • Use the fundamental identities to solve trigonometric equations. • Express trigonometric expressions in simplest form. • Solve trigonometric equations by factoring. • Solve trigonometric equations by using the Quadratic Formula. Thales of Miletus (circa 625–547 BC) is known as the founder of geometry. The legend is that he calculated the height of the similar triangles, which he developed by measuring the shadow of his staff. Based on proportions, this theory has applications in a number of areas, including fractal geometry, engineering, and architecture. Often, the angle of elevation and the angle of depression are found using similar triangles. Figure \(\PageIndex\): Egyptian pyramids standing near a modern city. (credit: Oisin Mulvihill) In earlier sections of this chapter, we looked at trigonometric identities. Identities are true for all values in the domain of the variable. In this section, we begin our study of trigonometric equations to study real-world scenarios such as the finding the dimensions of the pyramids. Solving Linear Trigonometric Equations in Sine and Cosine Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just...