qr code vb.net Primary Circular Functions in .NET framework

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Primary Circular Functions
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The sine function Look again at Fig 2-1 Imagine that ray OP points along the x axis, and then starts to rotate counterclockwise at steady speed around its end point O, as if that point is a mechanical bearing The point P, represented by coordinates (x0, y0), therefore revolves around O, following the unit circle Imagine what happens to the value of y0 (the ordinate of point P ) during one complete revolution of ray OP The ordinate of P starts out at y0 = 0, then increases until it reaches y0 = 1 after P has gone 1/4 of the way around the circle (that is, the ray has turned through an angle of p /2) After that, y0 begins to decrease, getting back to y0 = 0 when P has gone 1/2 of the way around the circle (the ray has turned through an angle of p ) As P continues in its orbit, y0 keeps decreasing until the value of y0 reaches its minimum of 1 when P has gone 3/4 of the way around the circle (the ray has turned through an angle of 3p /2) After that, the value of y0 rises again until, when P has gone completely around the circle, it returns to y0 = 0 for q = 2p The value of y0 is defined as the sine of the angle q The sine function is abbreviated as sin, so we can write
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sin q = y0
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Circular motion Imagine that you attach a glow-in-the-dark ball to the end of a string, and then swing the ball around and around at a steady rate of one revolution per second Suppose that you make the ball circle your head so the path of the ball lies in a horizontal plane Imagine that you are in the middle of a flat, open field at night The ball describes a circle as viewed from high above, as shown in Fig 2-2A If a friend stands far away with her eyes exactly in the plane of the ball s orbit, she sees a point of light that oscillates back and forth, from right-to-left and left-to-right, along what appears to be a straight-line path (Fig 2-2B) Starting from its rightmost apparent position, the glowing point moves toward the left for 1/2 second, speeding up and then slowing down; then it reverses direction; then it moves toward the right for
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Figure 2-2 Orbiting ball and string
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At A, as seen from above; at B, as seen edge-on
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A Fresh Look at Trigonometry
1/2 second, speeding up and then slowing down; then turns around again As seen by your friend, the ball reaches its extreme rightmost position at 1-second intervals, because its orbital speed is one revolution per second
The sine wave If you graph the apparent position of the ball as seen by your friend with respect to time, the result is a sine wave, which is a graphical plot of a sine function Some sine waves rise higher and lower (corresponding to a longer string), some are flatter (the equivalent of a shorter string), some are stretched out (a slower rate of revolution), and some are squashed (a faster rate of revolution) But the characteristic shape of the wave, known as a sinusoid, is the same in every case You can whirl the ball around faster or slower than one revolution per second, thereby altering the frequency of the sine wave: the number of times a complete wave cycle repeats within a specified interval on the independent-variable axis You can make the string longer or shorter, thereby adjusting the amplitude of the wave: the difference between the extreme values of its dependent variable No matter what changes you might make of this sort, the sinusoid can always be defined in terms of a moving point that orbits a central point at a constant speed in a perfect circle If we want to graph a sinusoid in the Cartesian plane, the circular-motion analogy can be stated as
y = a sin bq where a is a constant that depends on the radius of the circle, and b is a constant that depends on the revolution rate The angle q is expressed counterclockwise from the positive x axis Figure 2-3 illustrates a graph of the basic sine function; it s a sinusoid for which a = 1 and b = 1, and for which the angle is expressed in radians
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