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Trigonometry Lecture Notes_part1-1

8 undefine d notice how as we approach 90

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8 Undefine d **Notice how as we approach 90 ringoperator the graphs increases slowly at first then dramatically. This due to the vertical asymptote at 2 π . Below are some properties of the tangent function and its graph: Period: π Domain: All real numbers except π /2 + k π , where k an integer Range: All real numbers Symmetric with respect to the origin Vertical asymptotes at all odd multiples of π /2 Below is the graph of y = tan x:
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Graphing y = A tan( B x C )
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Example 30 Graph 2tan 2 x y = for 3 x π π - < < Find two consecutive asymptotes by solving 2 Bx C π - = - and 2 Bx C π - = . Identify an x-intercept midway between the asymptotes (average the asymptotes x’s). Find points ¼ and ¾ of the way between consecutive asymptotes. These points have y- coordinates –A and A. Connect these points with a smooth curve. Example 31 Graph two full periods of tan 4 y x π = + Find two consecutive asymptotes by solving 2 Bx C π - = - and 2 Bx C π - = . Identify an x-intercept midway between the asymptotes (average the asymptotes x’s). Find points ¼ and ¾ of the way between consecutive asymptotes. These points have y- coordinates –A and A. Connect these points with a smooth curve. The Cotangent Curve: The Graph of y = cotx and Its Characteristics
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Graphing Variations of y = cot x Graphing y=Acot(Bx-C) Example 32 Graph 3cot 2 y x = Find two consecutive asymptotes by solving 0 Bx C - = and Bx C π - = . Identify an x-intercept midway between the asymptotes (average the asymptotes x’s). Find points ¼ and ¾ of the way between consecutive asymptotes. These points have y- coordinates A and -A. Connect these points with a smooth curve.
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-3 -2 -1 1 2 3 -10 -8 -6 -4 -2 2 4 6 8 10 The Cosecant Curve: The Graph of y = cscx and Its Characteristics For the Cosecant and Secant curves it is helpful to graph the related Sine and Cosine curves first. X-intercepts on the sine curve correspond to vertical asymptotes on the cosecant curve. A maximum on the sine curve represents a minimum on the cosecant curve. A minimum on the sine curve represents a maximum on the cosecant curve.
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Example 33 The Secant Curve: The Graph of y=secx and Its Characteristics X-intercepts on the cosine curve correspond to vertical asymptotes on the secant curve. A maximum on the cosine curve represents a minimum on the secant curve. A minimum on the cosine curve represents a maximum on the secant curve.
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Example 34 Graph 3sec 2 x y = - for 5 x π π - < < (Hint: use the graph of 3cos 2 x y = - ) Section 7.6 Inverse Trigonometric Functions Recall that in order for a function to have an inverse it must pass the horizontal line test (i.e.-it must be one-to-one). Also, the inverse function’s graph will be the reflection of the original function about the line y = x. Since the trig functions would fail to be one-to-one on their entire domain due to their periodic nature among other things, we must restrict their domains to be able to find their inverses. The Inverse Sine Function The inverse sine function , denoted by sin -1 , is the inverse of the restricted sine function: y = sin x , - π /2 < x < π / 2 . Thus, y = sin -1 x means sin y = x , “at what angle is the sine equal to x?” where - π /2 < y < π /2 and –1 < x < 1. We read y = sin -1 x as “ y equals the inverse sine at x .”
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