chapter 8: trigonometric functions and applications

Copyright © 2007 Pearson Education, Inc. Slide 8-2

Chapter 8: Trigonometric Functions And Applications

8.1 Angles and Their Measures

8.2 Trigonometric Functions and Fundamental Identities

8.3 Evaluating Trigonometric Functions

8.4 Applications of Right Triangles

8.5 The Circular Functions

8.6 Graphs of the Sine and Cosine Functions

8.7 Graphs of the Other Circular Functions

8.8 Harmonic Motion


8.7 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.

xsin xcsc


Precalculus 1 and 2

Agenda

Write the COSECANT values of ALL the reference angles from 0 to 2π. (remember cosecant is the INVERSE of sine!)

Sketch the cosecant values along an X-Y axis.

Repeat with Secant…!

SWBAT:

• Sketch the cosecant graph

• Sketch the secant graph

• Recognize the

• HW: MCAS area and volume review…


Precalculus 1 and 2

Agenda




Remember TAN=sine/cosine

SWBAT:



• Recognize that we have an exam tomorrow!!! You need to know…

• HW: Study!


Advanced Algebra All Stars

Agenda: May 6th, 2011

Do Now – Last page in packet – please hand in for credit!

Class work-IN PAIRS, work on MCAS packet. You will be challenged-meet the challenge!!!

….if done….start

HW: Reflections Worksheet!

SWBAT:

• Welcome our guest

• Work efficiently and effectively with partner to complete the MCAS Challenge packet!!!

• Tackle problems as a TEAM!


Precalculus 2

AgendaDo Now – PEMDAS again. Yeah, you

should know it!




Remember TAN=sine/cosine

HW: Work on Portfolio!!

SWBAT:

• Identify transformations in Sine and Cosine Graphs



• Identify what COSECANT and SECANT graphs look like and explain WHY!


8.7 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.


8.7 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.– No x-intercepts.– Its period is 2 with no amplitude.– Symmetric with respect to the y-axis, and is an even

function.

2


8.7 Sketching Traditional Graphs of the Cosecant and Secant Functions

To graph y = a csc bx or y = a sec bx, with b > 0,

1. Graph the corresponding reciprocal function as a guide, using a dashed curve.

2. Sketch the vertical asymptotes. They will have equations of the form x = k, k an x-intercept of the guide function.

3. 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 bxy = a sec bx y = a cos bx


8.7 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.

.21

sec2 xy

.21

cos2 xy

].4,0[or,221

0 x


8.7 Graphing y = a sec bx

Sketch vertical asymptotes where the guide function equals 0 and draw the U-shaped branches, approaching the asymptotes.


8.7 Graphs of Tangent and Cotangent Functions

• Tangent– 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.

,22 to


8.7 Graphs of Tangent and Cotangent Functions

• Cotangent– 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) .2


8.7 Sketching Traditional Graphs of the Tangent and Cotangent Functions

To graph y = a tan bx or y = a cot bx, with b > 0,

1. The period is To locate two adjacent vertical asymptotes, solve the following equations for x:

2. Sketch the two vertical asymptotes found in Step 1.3. Divide the interval formed by the vertical asymptotes into

four equal parts.4. Evaluate the function for the first-quarter point, midpoint,

and third-quarter point, using x-values from Step 3.5. Join the points with a smooth curve approaching the

vertical asymptotes.

.b

For y = a tan bx: bx = and bx =

For y = a cot bx: bx = 0 and bx = .2 .2


8.7 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 intervalinto four equal parts and findingthe key points, we get

.2cot21

xy

.2

2

.002

xx

xx

20 x

.21

,8

3,0,

4,

21

,8

chapter 8: trigonometric functions and applications

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