This document discusses graphs of tangent, cotangent, secant, and cosecant functions. It provides information on asymptotes and zeros of these functions. An example problem demonstrates solving a trigonometric equation graphically by finding the intersection of the graphs of y = csc(x) and y = x^2. A quick review section tests understanding of period, zeros, and asymptotes of various trigonometric functions.

1. 4.5
Graphs of
Tangent,
Cotangent,
Secant, and
Cosecant
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2. What you’ll learn about
 The Tangent Function
 The Cotangent Function
 The Secant Function
 The Cosecant Function
… and why
This will give us functions for the remaining
trigonometric ratios.
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3. Asymptotes of the Tangent Function

2
3
2

3
2


2
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4. Zeros of the Tangent Function
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5. Asymptotes of the Cotangent Function
  0
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6. Zeros of the Cotangent Function

2
3
2

3
2


2
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7. The Secant Function

2
3
2

3
2


2
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8. The Cosecant Function
 0 
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9. Basic Trigonometry Functions
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10. Basic Trigonometry Functions
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11. Example Solving a Trigonometric
Equation Graphically
Find the smallest possible number x such that x2  csc x.
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12. Example Solving a Trigonometric
Equation Graphically
Find the smallest possible number x such that x2  csc x.
The graphs of y  2  x2 and
y  sec x are shown.
Using the grapher, we find
that the smallest positive
x-coordinate where the graphs
intersect is x  0.7749
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13. Quick Review
State the period of the function.
1. y  cos 4x
2. y  sin
1
4
x
Find the zeros and the vertical asymptotes of the function.
3. y 
x 1
x 1
4. y 
x 1
x  2x  3
5. Tell whether y  x2  4 is odd, even, or neither.
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14. Quick Review Solutions
State the period of the function.
1. y  cos 4x  /2
2. y  sin
1
4
x 8
Find the zeros and the vertical asymptotes of the function.
3. y 
x 1
x 1
1; x  1
4. y 
x 1
x  2x  3
1; x  3, x  2
5. Tell whether y  x2  4 is odd, even, or neither. even
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