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Study Guide for Exam 3 MAT170
Section 6.1 – Introduction to periodic functions
 Understand the concept of a function that repeats its output on a regular basis (the period). The ferris wheel
example on pages 232-235 is good. The period is summarized on page 235.
 Know how to identify the midline, amplitude and period given a graph or given general information about the
function.
Section 6.2 – The sine and cosine functions
 Understand the protocol of measuring angles (the unit circle, etc., page 240). You should be able to recognize 0,
45, 90, 180, 270 and 360 by sight.
 Know what it means to calculate a very large angle and negative angles.
 Know the definition of sin  and cos , page 241 and how to find the coordinates of a point determined by an angle
 on a circle of radius r (box on page 243: x= r cos  , y= r sin )
 Know the values of sin  and cos  for the cardinal angles.
Section 6.3- Radians
The definition of a radian is given on page 247. It is fundamentally related to the arc-length of the unit circle.
 Know how to convert back and forth between radians and degrees (page 248).
 Know the formula for the arclength and its applications (see Examples 4 and 5)
Section 6.4 – Graphs of the sine and cosine functions.
 Know the values for sin t and cos t for 30, 45 and 60 angles and how to derive values for all angles integer
multiples of these (e.g. 120, 210, 315 etc) . Remember: 30=/6 rads, 45= /4 rads, 60=/3 rads, etc.. These
values can be found on page 253. Know the EXACT values (with square roots).
 Know the sine and cosine function graphs, page 254.
 The cosine function is even, the sine function is odd. Sine and Cosine functions are periodic of period 2.
Section 6.5 – Sinusoidal functions
 Know the general form of a sinusoidal function, bottom of page 261.
 Know the meaning of A, B, h, k and how to find them graphically and from a word problem.
 Know that h is the horizontal shift while Bh is the phase shift and that period =2/|B|.
 Know how to find a formula for a sinusoidal function given the graph (see Example 2)
 See Examples 7 and 8, page 263 for good examples of constructing general sine/cosine functions from a
descriptive word problem. Suggestion: sketch a quick graph to see midline, amplitude and period.
Section 6.6 – Other trigonometric functions
 Know the tangent (tan t) function on pages 268-270. Remember: tan t = (sin t)/(cos t) = slope of ray from the origin
of angle t.
 Know the fundamental identity relating sine and cosine (page 270) . See Example 4 for an application.
 Know the definition and the graphs of the reciprocal of the trigonometric functions: sec t, csc t, cot t.
Section 6.7 – Inverse trigonometric functions

Know the definitions of inverse sine, cosine and tangent functions. Think of sin
whose sine is x. Think of cos
[

 
1
1
x as the angle in [ 
, ]
2 2
x as the angle in [0,] whose cosine is x. Think of tan 1 x as the angle in
, ] whose tangent is x.
2 2
KNOW YOUR BASIC VALUES FOR SINE, COSINE AND TANGENT IN EXACT FORM! E.g.
sin 1 (1 / 2)   / 6 radians (Examples 4,5,6).



 
Know the domain and range of the inverse trigonometric functions.
Arcsine and arctangent are odd functions, arccosine is neither even nor odd.
Know how to solve a trigonometric equation graphically, Example 1, page 273 and Example 2, page 274.


Know how to use the inverse sine, cosine and tangent functions to solve trigonometric equations. Remember, these
functions will only give one value back if you use your calculator, so you will need to use your knowledge of the
sine, cosine and tangent graphs to determine the other solutions, if they are applicable. See Examples 3, 10, 11, 12.
Know how to use reference angles.
Go over all the examples in this section.
Section 7.1 – Right triangles
Study the relationship between the trigonometric functions and right triangles, box on page 288. Study all the examples
in this section and the assigned hw problems.
Section 7.2 – Non-right triangles: Law of Sines and Cosines
Know the Law of Cosines (bottom of page 293) and the Law of Sines (bottom of page 295).
Study all examples in this section and the assigned hw problems.
Section 7.3 – Trigonometric identities
 Know the basic trig identities given on page 304.
 Know how to solve trig equations using identities, examples 1, 2 and 3, pages 300-3.
 Know how to verify (prove) an identity.
Strategies for proving an identity:
1. Begin with the more complicated expression and work toward the less complicated expression.
2. If no other move suggests itself, convert the entire expression to one involving sine and cosine.
3. combine fractions by writing them over a common denominator.
(a  b)(a  b)  a 2  b 2 to set up applications of the Pythagorean identities (e.g.
(1  sin  )(1  sin  )  1  sin 2   cos 2  )
4.
use the algebraic identity
5.
always be mindful of the “target” expression and favor manipulation that brings you closer to your goal.
Study all the assigned hw problems.
Section 7.4 – Sum and difference formulas for sine and cosine
 Know how to use the sum of angles and difference of angles formulas for sine and cosine (box at the bottom of
page 308) to find exact values of angles (see Example 1) and to prove simple identities relating sine and cosine (see
hw problem 8).
 Know how to use the sum and difference formulas for sine and cosine (page 309) to solve equations involving
sum/difference of sine or sum/difference of cosine functions (see hw problem 14).
The formulas in this section will be given in the test.
Section 7.6 – Polar coordinates
 Understand the relation between cartesian and polar coordinates (page 323).
 Know how to convert from polar to rectangular coordinates and vice versa. When going from rectangular
coordinates to polar be especially careful when determining the angle . In general   tan ( y / x) . You need
to plot the point first and determine in which quadrant it lies in order to find the correct value of .
 Know how to sketch the graph of simple polar equations (see Examples 2 and 3 and hw 12).
 Know how to describe regions in the plane using inequalities in polar coordinates (examples 4 and 5).
Go over all the examples in this section.
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