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TRIG-Fall 2008-Morrison
Trigonometry, 7th edition, Lial/Hornsby/Schneider, Addison Wesley, 2001
Chapter 1 Lecture Notes
Section 1.2
I.
Angles
Basic Terms
A.
Two distinct points determine a line called line AB.
B.
Line segment AB—a portion of the line between A and B, including points A and B.
C.
Ray AB—portion of line AB that starts at A and continues through B, and on past B.
D.
E.
F.
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.
G.
H.
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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II.
Types of Angles
The most common unit for measuring angles is the degree.
III.
Complementary & Supplementary Angles
A.
If the sum of the measures of two positive angles is 90˚, the angles are called
complementary.
B.
Two positive angles with measures whose sum is 180˚ are supplementary.
Example 1
IV.
Find the complement and the supplement of the angle with measure 18˚.
Degrees, Minutes, Seconds
A.
One minute is 1/60 of a degree.
B.
One second is 1/60 of a minute or 1/3600 of a degree.
C.
D.
Make sure your calculator is in Degree mode.
The DMS feature on the graphing calculator is found under 2nd ANGLE.
Example 2
Convert the angle measure of 36.624˚ to degrees, minutes, and seconds.
Example 3
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
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V.
Standard Position
A.
An angle is in standard position if its vertex is at the origin and its initial side is along
the positive x-axis.
B.
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.
VI.
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. Such angles are called coterminal
angles. The rotation can also be in the negative direction.
Example 4
VII.
Application
Example 5
Find the angles of smallest possible positive measure coterminal with each
angle.
a) 1115˚
b) -187˚
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
Angle Relationships and Similar Triangle
The mathematical definition of vertical angles is: two angles whose sides form pairs of
opposite rays. Vertical angles have equal measures.
1 and 2 is a pair of vertical angles.
Parallel lines are lines that lie in the same plane and do not intersect.
When a line intersects two parallel lines, it is called a transversal.
The transversal intersecting the parallel lines forms eight angles.
Special Properties:
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
alternate interior angles are congruent.
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
consecutive interior angles are supplementary.
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of
alternate exterior angles are congruent.
Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines,
then it is perpendicular to the second.
Corresponding Angles
Angle measures are equal.
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Example – A road crosses a set of railroad tracks. If the measure of  6 is
110o, find m  3.
Examples: Find the values of x and y
1.
6
2.
3.
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Triangles:
The sum of the measures of the angles of any triangle is 180°.
Types of Triangles
Right Triangles
A right triangle is a triangle with a right angle (i.e. 90°). You may have noticed that the side opposite the right angle is
always the triangle's longest side. It is called the hypotenuse of the triangle. The other two sides are called the legs. The
lengths of the sides of a right triangle are related by the Pythagorean Theorem. There are also special right triangles.
Acute Triangles
An acute triangle is a triangle whose angles are all acute (i.e. less than 90°). In the acute triangle shown below, a, b and
c are all acute angles.
Obtuse Triangles
An obtuse triangle has one obtuse angle (i.e. greater than 90º). The longest side called the hypotenuse is always
opposite the obtuse angle. In the obtuse triangle shown below, a is the obtuse angle.
Equilateral Triangles
An equilateral triangle has all three sides equal in length. Its three angles are also equal and they are each 60º.
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Isosceles Triangles
An isosceles triangle has two sides of equal length. The angles opposite the equal sides are also equal.
Scalene Triangles
A scalene triangle has no sides of equal length. Its angles are also all different in size.
Example: Two angles in a triangle are 63.6° and 42.1°. Determine the third angle.
Solution:
Let x be the third angle which we are to determine. Then, since all three angles must add up to
1800, we must have that
x + 63.6 + 42.1 = 180.
So
x = 180 – (63.6 + 42.1) = 74.3.
The third angle must be 74.3°.
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Similar and Congruent Triangles
Similar Triangles are triangles of exactly the same shape but not necessarily the same size.
Congruent triangles have both the same size and same shape. Triangles are similar if their
corresponding (matching) angles are equal and the ratios of their corresponding sides are in
proportion.
Two possible answers:
Small triangle on top:
Large triangle on top:
x = 20
x = 20
Find BE:
Use FULL sides of the triangles, cross
multiply and solve.
4x + 36 = 12x
36 = 8x
4.5 = x
Use the rule related to parallel lines, cross
multiply and solve.
36 = 8x
4.5 = x
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Example
A research team wishes to determine the altitude of a mountain as follows: They use a light source at L,
mounted on a structure of height 2 meters, to shine a beam of light through the top of a pole P' through
the top of the mountain M'. The height of the pole is 20 meters. The distance between the altitude of the
mountain and the pole is 1000 meters. The distance between the pole and the laser is 10 meters. We
assume that the light source mount, the pole and the altitude of the mountain are in the same plane. Find
the altitude h of the mountain.
We first draw a horizontal line LM. PP' and MM' are vertical to the ground and
therefore parallel to each other. Since PP' and MM' are parallel, the triangles LPP'
and LMM' are similar. Hence the proportionality of the sides gives:
1010 / 10 = (h - 2) / 18

Solve for h to obtain
h = 1820 meters
Example: Hank needs to determine the distance AB across a lake in an east-west
direction as shown in the illustration to the right. He can’t measure this distance directly
over the water. So, instead, he sets up a situation as shown. He selects the point D from
where a straight line to point B stays on land so he can measure distances. He drives a
marker stick into the ground at another point C on the line between points D and B. He
then moves eastward from point D to point E, so that the line of sight from point E to
point A includes the marker stick at point C.
Finally, with a long measuring tape, he determines that
DE = 412 m
DC = 260 m
BC = 1264 m
and
CE = 308 m
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Determine if this is enough information to calculate the distance AB, and if so, carry
out the calculation.
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Section 1.4
I.
Definitions of the Trigonometric Functions
Definition 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
d2
i
3,2 . Find the values of the six trigonometric functions of angle θ.
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Example 3
given
ratio is positive or negative:
Example 4
II.
Suppose that point (x, y) is in the indicated quadrant. Decide whether the
IV,
y
r
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.
Trigonometric Function Values of Quadrantal Angles
Memorize at least the sine and cosine values in this table. Memorizing the tangent values is
also helpful.
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 1.5
I.
Using the Definitions of the Trigonometric Functions
Reciprocal Identities
(These identities hold for any angle θ that does not lead to a 0 denominator.)
II.
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 Function Values
All Stores Take Cash or All Students Take Calculus
Example 3
Identify the quadrant (or quadrants) for any angle that satisfies tan θ > 0,
cot θ > 0
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III.
Ranges of Trigonometric Functions
For any angle θ for which the indicated functions exist:
A.
-1 ≤ sin θ ≤ 1
and
-1 ≤ cos θ ≤ 1;
B.
tan θ and cot θ can equal any real number;
C.
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.)
Example 4
IV.
Decide whether the following statement is possible or impossible for an angle θ:
cos θ = -1.001
Pythagorean Identities
V.
Quotient Identities
Example 5
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.