Mathematics
Translations of Trigonometric Functions
Translations of trigonometric functions refer to the shifting of the graphs of sine, cosine, and tangent functions along the x-axis or y-axis. This shifting is achieved by adding or subtracting values within the function, resulting in a change in the phase or period of the graph. Understanding these translations is crucial for analyzing and graphing trigonometric functions.
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6 Key excerpts on "Translations of Trigonometric Functions"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 3 Trigonometric Functions In mathematics, the trigonometric functions (also called circular functions ) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x -axis, the sine of the angle gives the length of the y -component (rise) of the triangle, the cosine gives the length of the x -component (run), and the tangent function gives the slope ( y -component divided by the x -component). More pr ecise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used for instance in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 10 Trigonometric Functions In mathematics, the trigonometric functions (also called circular functions ) are fun-ctions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x -axis, the sine of the angle gives the length of the y -component (rise) of the triangle, the cosine gives the length of the x -component (run), and the tangent function gives the slope ( y -component divided by the x -comp-onent). More precise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used for instance in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Trigonometric Functions In mathematics, the trigonometric functions (also called circular functions ) are func-tions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications. The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle with radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x -axis, the sine of the angle gives the length of the y -component (rise) of the triangle, the cosine gives the length of the x -component (run), and the tangent function gives the slope ( y -component divided by the x -component). More precise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used for instance in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.
eBook - PDF- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
2 Trigonometric Functions IN THIS PART . . . Define the basic trig functions using the lengths of the sides of a right triangle. Determine the relationships between the trig cofunctions and their shared sides. Extend your scope to angles greater than 90 degrees using the unit circle. Investigate the ins and outs of the domains and ranges of the six trig functions. Use reference angles to compute trig functions. Apply trig functions to real-world problems. CHAPTER 6 Describing Trig Functions 91 Chapter 6 Describing Trig Functions B y taking the lengths of the sides of right triangles or the chords of circles and creating ratios with those numbers and variables, our ancestors initi- ated the birth of trigonometric functions. These functions are of infinite value, because they allow you to use the stars to navigate and to build bridges that won’t fall. If you’re not into navigating a boat or engineering, then you can use the trig functions at home to plan that new addition. And they’re a staple for students going into calculus. You may be asking, “What is a function? What does it have to do with trigonom- etry?” In mathematics, a function is a mechanism that takes a value you input into it and churns out an answer, called the output. A function is connected to rules involving mathematical operations or processes. The six trig functions require one thing of you — inputting an angle measure — and then they output a number. These outputs are always real numbers, from infinitely small to infinitely large and everything in between. The results you get depend on which function you use. Although in earlier times, some of the function computations were rather tedious, today’s hand-held calculators, and even phones, make everything much easier. IN THIS CHAPTER » Understanding the three basic trig functions » Building on the basics: The reciprocal functions » Recognizing the angles that give the cleanest trig results » Determining the exact values of functions
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
542 chapter 9 Trigonometric Functions 9.4 More Trigonometric Functions learning objectives By the end of this section you should be able to evaluate the tangent of any multiple of 30 ◦ or 45 ◦ ( π 6 radians or π 4 radians); find the equation of the line making a given angle with the positive horizontal axis and containing a given point; sketch a radius of the unit circle corresponding to a given value of the tangent function; compute cos θ, sin θ, and tan θ if given just one of these quantities and the location of the corresponding radius; evaluate sec θ, csc θ, and cot θ as 1 divided by the value of one of the other trigonometric functions. Section 9.3 introduced the cosine and the sine, the two most important trigonometric functions. This section introduces the tangent, another key trigonometric function, along with three more trigonometric functions. Definition of Tangent Recall that cos θ and sin θ are defined to be the first and second coordinates of the endpoint of the radius of the unit circle corresponding to θ. The ratio of these two numbers, with the cosine in the denominator, turns out to be sufficiently useful to deserve its own name. Tangent The tangent of an angle θ, denoted tan θ, is defined by tan θ = sin θ cos θ provided that cos θ = 0. The radius of the unit circle corresponding to θ has its initial point at (0, 0) and its endpoint at (cos θ, sin θ). Thus the slope of this line segment Recall that the slope of the line segment connecting (x 1 , y 1 ) and (x 2 , y 2 ) is y 2 -y 1 x 2 -x 1 . equals sin θ-0 cos θ-0 , which equals sin θ cos θ , which equals tan θ. In other words, we have the following interpretation of the tangent of an angle: Tangent as slope tan θ is the slope of the radius of the unit circle corresponding to θ. The following figure illustrates how the cosine, sine, and tangent of an angle are defined: section 9.4 More Trigonometric Functions 543 Θ cos Θ, sin Θ slope tan Θ 1 The radius corresponding to θ has slope tan θ.
eBook - PDF- Geoffrey Berresford, Andrew Rockett(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 540 Chapter 8 Trigonometric Functions Similarly, cos u behaves just like the x -coordinate of the point ( x, y ) as it moves around the circle—first decreasing from 1 to 0 to 2 1, then increasing back to 1. Again, this cycle repeats itself each time the angle completes another full circle of 2 p radians, giving the following graph of cos t . Graph of cos t t 3 2 H11002 2 H9266 H9266 H9266 H9266 H9266 H9266 H9266 H9266 H9266 H11002 3 H9266 H11002 H11002 1 1 3 2 H11002 2 H11002 2 3 2 cos t Modified Sine and Cosine Functions y 5 a sin bt y 5 a cos bt f have amplitude a and period 2 p b . H11002 a a amplitude a period (one complete cycle) 2 H9266 b 2 H9266 b t y y H11005 a sin ( bt ) H11002 a a amplitude a 2 H9266 b t y y H11005 a cos ( bt ) period (one complete cycle) 2 H9266 b H9258 H9266 H9266 H9266 2 3 H9266 2 H11002 1 1 cos H9258 2 (These graphs could also have been found by the more tedious process of plotting points.) Sin t and cos t are said to be periodic with period 2 p , meaning that they repeat their values every 2 p units. We will now think of sin t and cos t as ordinary functions, unrelated to angles, functions that oscillate between 1 1 and 2 1 in a “wavy” pattern. We will use sin t and cos t to model quantities that undergo regular fluctuations, such as seasonal sales. Modified Sine and Cosine Curves—Changing Amplitude and Period The amplitude of a sine or cosine curve is half the distance between its highest and lowest values (sometimes called its half-height ).
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