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TRIG-Fall 2011-Jordan
Trigonometry, 9th edition, Lial/Hornsby/Schneider, Pearson, 2009
Chapter 1: Trigonometric Functions
Section 1.1
Angles
Basic Terms
Ray AB—portion of line AB that starts at A and continues through B, and on past B
Angle—formed by rotating a ray around its endpoint
The ray in its initial position is called the initial side of the angle.
The ray in its location after the rotation is the terminal side of the angle.
Positive angle: The rotation of the terminal side of an angle is counterclockwise.
Negative angle: The rotation of the terminal side is clockwise.
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Types of Angles
The most common unit for measuring angles is the degree.
Complementary & Supplementary Angles
If the sum of the measures of two positive angles is 90˚, the angles are called
complementary.
Two positive angles with measures whose sum is 180˚ are supplementary.
Example 1
Find the complement and the supplement of the angle with measure 18˚.
Example 2
Find the measure of each angle.
Degrees, Minutes, Seconds
One minute is 1/60 of a degree.
One second is 1/60 of a minute or 1/3600 of a degree.
Make sure your calculator is in Degree mode.
The DMS feature on the graphing calculator is found under 2nd ANGLE.
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Example 3
Convert the angle measure of 36.624˚ to degrees, minutes, and seconds.
Example 4 Convert the angle measure of 74˚ 12′ 18″ to decimal degrees. Round to
the nearest thousandth of a degree.
TI-83 Keystrokes: 74, 2nd ANGLE, ˚, 12, 2nd ANGLE, ′, 18, ALPHA +, ENTER
TI-82 Keystrokes: 74, 2nd ANGLE, ′, 12, 2nd ANGLE, ′, 18, 2nd ANGLE, ′, ENTER
Standard Position
An angle is in standard position if its vertex is at the origin and its initial side is along
the positive x-axis.
Angles in standard position having their terminal sides along the x-axis or y-axis, such
as angles with measures 90˚, 180˚, 270˚, and so on, are called quadrantal angles.
Coterminal Angles
A complete rotation of a ray results in an angle measuring 360˚. By continuing the
rotation, angles of measure larger than 360˚ can be produced. Angles whose measures
differ by multiples of 360˚ are called coterminal angles. The rotation can also be in the
negative direction.
Example 5
Find the angle of least positive measure coterminal with each angle.
a) 1115˚
b) -187˚
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Application
Example 6 An airplane propeller rotates 1000 times per minute. Find the number of
degrees that a point on the edge of the propeller will rotate in 1 second.
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Section 1.3
Trigonometric Functions
Definitions of Trigonometric Functions
Let (x, y) be a point other than the origin on the terminal side of an angle θ in standard
position. The distance from the point to the origin is r  x 2  y 2 .
The six trigonometric functions of θ are defined as follows:
Example 1 The terminal side of angle θ in standard position passes through the point
(12, 16). Find the values of the six trigonometric functions of angle θ.
Example 2
The terminal side of angle θ in standard position passes through the point
. Find the values of the six trigonometric functions of angle θ.
Example 3
Suppose that point (x, y) is in the indicated quadrant. Decide whether the
y
given ratio is positive or negative:
IV,
r
Example 4 The terminal side of angle θ in standard position passes through the point
(-4, 0). Find the values of the six trigonometric functions of angle θ.
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Trigonometric Function Values of Quadrantal Angles
Memorize at least the sine and cosine values in this table. Memorizing the tangent
values is also helpful.
This is easier to memorize if you use the figure below. Since any point on the terminal
side of the angle can be used to find the trig ratios, choose the point one unit from the
origin so that r = 1.
Example 5 Use the trigonometric function values of quadrantal angles to evaluate the
following expression:
2 sec 0˚ + 4 cot 2 90˚ + cos 360˚
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Section 1.4
Using the Definitions of the Trigonometric Functions
Reciprocal Identities
These identities hold for any angle θ that does not lead to a 0 denominator.
Example 1
Use the appropriate reciprocal identity to find cos θ if sec θ = 2/3
Example 2
Use the appropriate reciprocal identity to find sin θ if csc   
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Signs of Trigonometric Function Values
All Stores Take Cash or All Students Take Calculus
Example 3
cot θ > 0
Identify the quadrant (or quadrants) for any angle that satisfies tan θ > 0,
Example 4 Find the signs of the six trigonometric function values for the angle in
standard position whose measure is -60°.
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Ranges of Trigonometric Functions
For any angle θ for which the indicated functions exist:
-1 ≤ sin θ ≤ 1
and
-1 ≤ cos θ ≤ 1;
tan θ and cot θ can equal any real number;
sec θ ≤ -1 or sec θ ≥ 1
and
csc θ ≤ -1 or csc θ ≥ 1.
(Notice that sec θ and csc θ are never between -1 and 1.)
Pythagorean Identities
Example 5
Quotient Identities
Find csc θ, if cot θ = -1/2, with θ in quadrant IV
Example 6 Find all trigonometric function values for the angle θ if tan   3 , with θ in
quadrant III.