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Figure 2-3 Graph of the sine function for
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values of q between 3p and 3p Each division on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
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The cosine function Look again at Fig 2-1 Imagine, once again, a ray OP running outward from the origin through point P on the circle Imagine that at first, the ray points along the x axis, and then it rotates steadily in a counterclockwise direction Now let s think about what happens to the value of x0 (the abscissa of point P ) during one complete revolution of ray OP It starts out at x0 = 1, then decreases until it reaches x0 = 0 when q = p /2 Then x0 continues to decrease, getting down to x0 = 1 when q = p As P continues counterclockwise around the circle, x0 increases When q = 3p /2, we get back up to x0 = 0 After that, x0 increases further until, when P has gone completely around the circle, it returns to x0 = 1 for q = 2p The value of x0 is defined as the cosine of the angle q The cosine function is abbreviated as cos, so we can write
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The cosine wave Circular motion in the Cartesian plane can be defined in terms of the cosine function by means of the equation
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y = a cos 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, just as is the case with the sine function The angle q is measured or defined counterclockwise from the positive x axis, as always The shape of a cosine wave is exactly the same as the shape of a sine wave Both waves are sinusoids But the entire cosine wave is shifted to the left by 1/4 of a cycle with respect to the sine wave That works out to an angle of p /2 Figure 2-4 shows a graph of the basic cosine
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cos q 3 2 1 3p 3p q 1 2 3
Figure 2-4 Graph of the cosine function for
values of q between 3p and 3p Each division on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
A Fresh Look at Trigonometry
function; it s a cosine wave for which a = 1 and b = 1 Because the cosine wave in Fig 2-4 has the same frequency but a difference in horizontal position compared with the sine wave in Fig 2-3, the two waves are said to differ in phase For those of you who like fancy technical terms, a phase difference of 1/4 cycle (or p /2) is known in electrical engineering as phase quadrature
The tangent function Once again, refer to Fig 2-1 The tangent (abbreviated as tan) of an angle q can be defined using the same ray OP and the same point P = (x0,y0) as we use when we define the sine and cosine functions The definition is
tan q = y0 /x0 We ve seen that sin q = y0 and cos q = x0, so we can express the tangent function as tan q = sin q /cos q The tangent function is interesting because, unlike the sine and cosine functions, it blows up at certain values of q This is shown by a graph of the function (Fig 2-5) Whenever x0 = 0, the denominator of either quotient above becomes 0, so the tangent function is not defined for any angle q such that cos q = 0 This happens whenever q is a positive or negative odd-integer multiple of p /2
tan q 3 2 1
3p q
3p 1 2 3
Figure 2-5 Graph of the tangent function for values
of q between 3p and 3p Each division on the horizontal axis represents p /2 units Each division on the vertical axis represents 1/2 unit
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