vb.net generate qr code A Fresh Look at Trigonometry in Visual Studio .NET

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A Fresh Look at Trigonometry
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Trigonometry (or trig ) involves the relationships between angles and distances Traditional texts usually define the trigonometric functions of an angle as ratios between the lengths of the sides of a right triangle containing that angle If you ve done trigonometry with triangles, get ready for a new perspective!
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Circles in the Cartesian Plane
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In Cartesian xy coordinates, circles are represented by straightforward equations The equation for a particular circle depends on its radius, and also on the location of its center point
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The unit circle In trigonometry, we re interested in the circle whose center is at the origin and whose radius is 1 This is the simplest possible circle in the xy plane It s called the unit circle, and is represented by the equation
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x2 + y2 = 1 The unit circle gives us an elegant way to define the basic trigonometric functions That s why these functions are sometimes called the circular functions Before we get into the circular functions themselves, let s be sure we know how to define angles, which are the arguments (or inputs) of the trig functions
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Naming angles Mathematicians often use Greek letters to represent angles The italic, lowercase Greek letter theta is popular It looks like an italic numeral 0 with a horizontal line through it (q) When writing about two different angles, a second Greek letter is used along with q Most often, it s the italic, lowercase letter phi This character looks like an italic lowercase English letter o with a forward slash through it (f)
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A Fresh Look at Trigonometry
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Sometimes the italic, lowercase Greek letters alpha, beta, and gamma are used to represent angles These, respectively, look like the following symbols: a, b, and g When things get messy and there are a lot of angles to talk about, numeric subscripts may be used with Greek letters, so don t be surprised if you see text in which angles are denoted q1, q2, q3, and so on If you read enough mathematical papers, you ll eventually come across angles that are represented by other lowercase Greek letters Angle variables can also be represented by more familiar characters such as x, y, or z As long as we know the context and stay consistent in a given situation, it really doesn t matter what we call an angle
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Radian measure Imagine two rays pointing outward from the center of a circle Each ray intersects the circle at a point Suppose that the distance between these points, as measured along the arc of the circle, is equal to the radius of the circle In that case, the measure of the angle between the rays is one radian (1 rad) There are always 2p rad in a full circle, where p (the lowercase, non-italic Greek letter pi) stands for the ratio of a circle s circumference to its diameter The number p is irrational Its value is approximately 314159 Mathematicians prefer the radian as a standard unit of angular measure, and it s the unit we ll work with in this course It s common practice to omit the rad after an angle when we know that we re working with radians Based on that convention:
An angle of p /2 represents 1/4 of a circle An angle of p represents 1/2 of a circle An angle of 3p /2 represents 3/4 of a circle An angle of 2p represents a full circle
An acute angle has a measure of more than 0 but less than p /2, a right angle has a measure of exactly p /2, an obtuse angle has a measure of more than p /2 but less than p, a straight angle has a measure of exactly p, and a reflex angle has a measure of more than p but less than 2p
Degree measure The angular degree ( ), also called the degree of arc, is the unit of angular measure familiar to lay people One degree (1 ) is 1/360 of a full circle You probably know the following basic facts:
An angle of 90 represents 1/4 of a circle An angle of 180 represents 1/2 of a circle An angle of 270 represents 3/4 of a circle An angle of 360 represents a full circle
An acute angle has a measure of more than 0 but less than 90 , a right angle has a measure of exactly 90 , an obtuse angle has a measure of more than 90 but less than 180 , a straight angle has a measure of exactly 180 , and a reflex angle has a measure of more than 180 but less than 360
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