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Pre-calculus Notes: Chapter 5 – The Trigonometric Functions
Section 1 – Angles and Degree Measure
Use the word bank below to fill in the blanks below. You may use each term only once.
degree
vertex
negative angle
terminal side
positive angle
initial side
standard position
An angle may be generated by rotating one of two rays that share a
fixed endpoint known as a _____________________. One of the rays
is fixed to form the ____________________________ of the angle, and
the second ray rotates to form the ______________________________.

If the rotation is in a counterclockwise direction, the angle formed is a _____________________

If the rotation is clockwise, it is a ______________________________.
An angle with its vertex at the origin and its initial side along the positive x-axis is said to be in
_________________________________. In the figures below, all the angles are in standard position.
The most common unit used to measure angles is the _________________________. In order to obtain
a more accurate angle measure, Babylonians also measured angles in minutes and seconds. A degree
is subdivided into 60 equal parts known as minutes (1’), and the minute is subdivided into 60 equal
parts know as seconds (1’’).
Example 1
Change the angle measure to degrees, minutes, and seconds.
a.
15.735o
b.
329.125o
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Example 2
Rewrite as a decimal rounded to the nearest thousandth.
a.
39o 5’ 34’’
b.
35o 12’ 7’’
If the terminal side of an angle that is in standard position coincides with one of the axes, the angle is
called a quadrantal angle, as in the figures below.
A full rotation around a circle is 3600. Measures of more than 3600 represent multiple rotations.
Example 3
Give the angle measure represented by each rotation.
a.
9.5 rotations clockwise
b.
6.75 rotations counterclockwise
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Examine the table below to determine the definitions for coterminal and reference angles. It may
help to sketch each angle.
Coterminal Angles
Reference Angles
o
o
Examples for 60
Non-Examples for 60
Example for 60o
Non-Examples for 60o
420o
120o
60o
120o
780o
-60o
420o
-300o
30o
780o
-660o
60o
-300o
Examples for 135o
495o
855o
-225o
-585o
Examples for 210o
570o
930o
-150o
-510o
Examples for -20o
340o
700o
-380o
-740o
Non-Examples for 135o
135o
45o
-135o
225o
Non-Examples for 210o
30o
210o
150o
-210o
Non-Examples for -20o
20o
-340o
-20o
340o
Example for 135o
45o
Example for 210o
30o
Example for -20o
20o
Non-Examples for 135o
-135o
225o
-225o
-45o
Non-Examples for 210o
-150o
150o
210o
-210o
Non-Examples for -20o
-20o
340o
-340o
Coterminal Angles: ________________________________________________________________________
Reference Angles: _________________________________________________________________________
Example 4
Identify all angles that are coterminal with each angle. Then find one positive angle and one
negative angle that are coterminal with the angle.
a.
86o
b.
294o
Example 5
If each angle is in standard position, determine a coterminal angle that is between 0o and 360o. State
the quadrant in which the terminal side lies.
a.
595o
b.
-777o
3
Example 6
Find the measure of the reference angle for each angle.
a.
1200
b.
-1350
c.
3120
d.
-1950
Section 2 – Trigonometric Ratios in Right Triangles
Example 1
Find the values of sine, cosine, and tangent for A .
Example 2
a.
If sec  
6
, find cos  .
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b.
If sin   0.8 , find csc .
4
Example 3
Find the values of the six trigonometric ratios for E .
Section 3 – Trigonometric Functions on the Unit Circle
cos   x
sec  
1
x
sin   y
tan  
y
x
1
y
cot  
x
y
csc  
Be cautious: division by zero is undefined, so there are
values of tangent, cotangent, secant, and cosecant that are
undefined.
Example 1
Use the unit circle to find each value.
a.
sin(-900)
b.
cot 2700
c.
sec 900
d.
cos(-1800)
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Example 2
Use the unit circle to find the values of the six trigonometric functions for a 2100 angle.
Example 3
Find the values of the six trigonometric functions for angle  in standard position if a point with the
coordinates (-15, 20) lies on its terminal side.
Example 4
4
Suppose  is an angle in standard position whose terminal side lies in Quadrant III. If sin    ,
5
find the values of the remaining five trigonometric functions of  .
Example 5
Suppose  is an angle in standard position whose terminal side lies in Quadrant IV. If sec  
29
,
5
find the values of the remaining five trigonometric functions of  .
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Section 4 – Applying Trigonometric Functions
Example 1
If J = 500 and j = 12, find r.
Example 2
The chair lift at a ski resort rises at an angle of 20.750 and attains a vertical height of 1200 feet.
a.
How far does the chair lift travel up the side of the mountain?
b.
A film crew in a helicopter records an overhead view of a skier’s downhill run from where she
gets off the chair lift at the top to where she gets back on the chair lift for her next run. If the
helicopter follows a level flight path, what is the length of that path?
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Example 3
A regular hexagon is inscribed in a circle with diameter 26.6 centimeters. Find the apothem of the
hexagon. (Apothem = the measure of the line segment from the center of the polygon to the midpoint
of one of its sides)
Angle of Elevation ________________________________________________________________________
Angle of Depression _______________________________________________________________________
Example 4
An observer in the top of a lighthouse determines that the angles of depression to two sailboats
directly in line with the lighthouse are 3.50 and 5.750, If the observer is 125 feet above sea level, find
the distance between the boats.
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Section 5.5 – Solving Right Triangles
Example 1
Solve each equation.
a.
tan x = 1
b.
sin x  
1
2
Example 2
Evaluate each expression. Assume that all angles are in Quadrant I.
a.
2

cos arccos 
5

b.
4

tan cos 1 
5

c.
2

cos arcsin 
3

Example 3
If g = 28 and h = 21, find H.
Example 4
Many cities place restrictions on the height and placement of skyscrapers in order to protect residents
from completely shaded streets. If a 100-foot building casts an 88-foot shadow, what is the angle of
elevation of the sun?
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Example 5
Solve each triangle described, given the triangle below. Round to the nearest tenth.
a.
K = 400, k = 26
b.
j = 65, l = 55
Section 5.6 – The Law of Sines
Example 1
Solve LMN if L = 290, M = 1120, and l = 22.
Example 2
A person in a hot-air balloon observes that the angle of depression to a building on the ground is
65.80. After ascending vertically 500 feet, the person now observes that the angle of depression is
70.20. How far is the balloonist now from the building?
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Example 3
Find the area of ABC if a = 4.7, c = 12.4, and B = 47020’.
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Section 7 – The Ambiguous Case for the Law of Sines
Example 1
Determine the number of possible solutions for each triangle.
a.
A = 300, a = 8, b = 10
b.
b = 8, c = 10, B = 1180
c.
A = 630, a = 18, b = 25
d.
A = 1050, a = 73, b = 55
Example 2
Find all solutions for each triangle. If no solutions exist, write ‘none’.
a.
A = 980, a = 39, b = 22
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b.
A = 72.20, a = 21, b = 22
Example 3
A group of contractors is constructing a 24-foot slide on a playground. The slide inclines 450 from the
horizontal. The access ladder measures 18 feet long. At what angle to the horizontal should the
contractors build the ladder?
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Section 8 – The Law of Cosines
Example 1
Suppose you want to fence a triangular lot. If two sides measure 84 feet and 78 feet and the angle
between the two sides is 1020, what is the length of the fence to the nearest foot?
Example 2
Solve each triangle.
a.
A = 39.40, b = 12, c = 14
b.
a = 19, b = 24.3, c = 21.8
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Example 3
Find the area of ABC if a = 24, b = 52, and c = 39.
Example 4
Find the area of ABC . Round to the nearest tenth.
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