2. Chapter 8: The Unit Circle and the Functions of Trigonometry
8.1 Angles, Arcs, and Their Measures
8.2 The Unit Circle and Its Functions
8.3 Graphs of the Sine and Cosine Functions
8.4 Graphs of the Other Circular Functions
8.5 Functions of Angles and Fundamental Identities
8.6 Evaluating Trigonometric Functions
8.7 Applications of Right Triangles
8.8 Harmonic Motion

3. 8.4 Graphs of the Other Trigonometric Functions
Graphs of the Cosecant and Secant Functions
Cosecant values are reciprocals of the corresponding sine values.
If sin x = 1, the value of csc x is 1. Similarly, if sin x = –1,
then csc x = –1.
When 0 < sin x < 1, then csc x > 1. Similarly,
if –1 < sin x < 0, then csc x < –1.
When approaches 0, the gets larger. The graph of y = csc x approaches the vertical line x = 0.
In fact, the vertical asymptotes are the lines x = n.

4. 8.4 Graphs of the Cosecant and Secant Functions
A similar analysis for the secant function can be done. Plotting a few points, we have the solid lines representing the curves for the cosecant and secant functions.

5. 8.4 Graphs of the Cosecant and Secant Functions
Cosecant Function
Discontinuous at values of x of the form x = n, and has vertical asymptotes at these values.
No x-intercepts.
Its period is 2 with no amplitude.
Symmetric with respect to the origin, and is an odd function.
Secant Function
Discontinuous at values of x of the form (2n + 1) , and has vertical asymptotes at these values.
Symmetric with respect to the y-axis, and is an even function.

6. 8. 4. Guidelines for Sketching Graphs of
8.4 Guidelines for Sketching Graphs of the Cosecant and Secant Functions
To graph y = a csc bx or y = a sec bx, with b > 0,
Graph the corresponding reciprocal function as a guide, using a dashed curve.
Sketch the vertical asymptotes. They will have equations of the form x = k, k an x-intercept of the guide function.
Sketch the graph of the desired function by drawing the
U-shaped branches between adjacent asymptotes.
To Graph
Use as a Guide
y = a csc bx
y = a sin bx
y = a sec bx
y = a cos bx

7. 8.4 Graphing y = a sec bx Example Graph Solution The guide function is
One period of the graph lies along the interval that
satisfies the inequality
Dividing this interval into four equal parts gives the
key points (0,2), (,0), (2,–2), (3,0), and (4,2),
which are joined with a smooth dashed curve.

8. 8.4 Graphing y = a sec bx
Sketch vertical asymptotes where the guide function
equals 0 and draw the U-shaped branches,
approaching the asymptotes.

9. 8.4 Graphs of Tangent and Cotangent Functions
Its period is  and it has no amplitude.
Its values are 0 when sine values are 0, and undefined when cosine values are 0.
As x goes from tangent values go from – to , and increase throughout the interval.
The x-intercepts are of the form x = n.

10. 8.4 Graphs of Tangent and Cotangent Functions
Its period is  and it has no amplitude.
Its values are 0 when cosine values are 0, and undefined when sine values are 0.
As x goes from 0 to , cotangent values go from  to
–, and decrease throughout the interval.
The x-intercepts are of the form x = (2n + 1)

11. 8. 4. Guidelines for Sketching Graphs of
8.4 Guidelines for Sketching Graphs of the Tangent and Cotangent Functions
To graph y = a tan bx or y = a cot bx, with b > 0,
The period is To locate two adjacent vertical asymptotes, solve the following equations for x:
Sketch the two vertical asymptotes found in Step 1.
Divide the interval formed by the vertical asymptotes into four equal parts.
Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using x-values from Step 3.
Join the points with a smooth curve approaching the vertical asymptotes.
For y = a tan bx: bx = and bx =
For y = a cot bx: bx = 0 and bx = .

12. 8.4 Graphing y = a cot bx Example Graph
Solution Since the function involves cotangent, we can
locate two adjacent asymptotes by solving the equations:
Dividing the interval
into four equal parts and finding
the key points, we get