Chapter 5 – The Trigonometric Functions

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Chapter 5 The Trigonometric Functions
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5.1 Angles and Their Measure
What is the Initial Side? And Terminal
Side? What are radians compared to
degrees? 1 radian degrees or
about 57.3o 1 degree radians or
about 0.017 radians
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Change 30o to radians
Change radians
to degrees.
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If a is the degree measure of an angle, then all
angles of the form a 360ko, where k is an
integer, are coterminal with a. If b is the
radian measure of an angle, then all angles of
the form b 2k , where k is an integer, are
coterminal with b. EX Find one positive angle
and one negative angle that are coterminal with
. EX Identify all angles that are
coterminal with a 60o angle.
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• Reference Angle Rule
• For any angle its
  reference angle is defined by
• when the terminal side is quadrant I
• when the terminal side is quadrant II
• when the terminal side is in quadrant III
• when the terminal side is in quadrant IV
• Ex Find the measure of the reference angle for
  each.
• 510o

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5.2 Central Angles and Arcs
A central angle of a circle is an angle whose
vertex lies at the center of the circle. Note
If two central angles in different circles are
congruent then the ratio of the length of their
intercepted arcs is equal to the ratio of the
measures of their radii. The length of any
circular arc, s, is equal to the product of the
measure of the radius of the circle, r, and the
radian measure of the central angle, , that
it subtends.
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Find the length of an arc that subtends a central
angle of 42o in a circle with radius of 8cm. If
an object moves along a circle of radius r units,
then its linear velocity, v, is given by Where
is the angular velocity in radians per unit
of time. A pulley of radius 12cm turns at 7
revolutions per second. What is the linear
velocity of the belt driving the pulley in meters
per second? A trucker drives 55 mph. His
trucks tires have a diameter of 26 inches. What
is the angular velocity of the wheels in
revolutions per second?
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If is the measure of the central angle
expressed in radians and r is the measure of the
radius of the circle, then the area of the
sector, A, is as follows A sector has arc
length of 16cm and a central angle measuring 0.95
radians. Find the radius of the circle and the
area of the sector.
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5.3 Circular Functions
Def If the terminal side of an angle in
standard position intersects the unit circle at
P(x,y), then and Find each
value
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For any angle in standard position with measure
, a point P(x,y) on its terminal side, and
, the sine and cosine functions of
are as follows Find the values of the sine
and cosine function of an angle in standard
position with measure if the point with
coordinates (3, 4) lies on its terminal
side. Find when and the
terminal side of is in the first quadrant.
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For any angle in standard position with measure
, a point P(x,y) on its terminal side, and
, the trigonometric functions of are
as follows The terminal side of an angle
in standard position contains the point with
coordinates (8, -15). Find tangent, cotangent,
secant, and cosecant for .
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If and lies in quadrant III,
find sine, cosine, tangent, cotangent, and secant
for .
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5.4 Trigonometric Functions of Special Angles
Lets first discuss the value of each of the trig
functions at Then lets create two very
special triangles which we will then use to help
find many other trig values.
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5.5 Right Triangles
For an acute angle A in a right triangle ABC, the
trigonometric functions are as follows
B
c
a
A
C
b
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For the following triangle find the values of the
six trig functions of A. Solve right
triangle ABC. Round angle measures to the
nearest degree and side measures to the nearest
tenth. A 49o
13
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A
12
A
c
b
B
7
C
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• In the given triangle find the measure of angle R
  to the nearest degree.
• Assume that a ladder is mounted 8ft off the
  ground.
• How far from an 84ft burning building should the
  base of the ladder be placed to achieve the
  optimum operating angle of 60o?
• How far should the ladder be extended to reach
  the roof?

T
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8
R
S
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A flagpole 40ft high stands on top of the
Wentworth Building. From a point P in front of
Baileys Drugstore, the angle of elevation of the
top of the pole is 54o54, and the angle of
elevation of the bottom of the pole is 47o30.
To the nearest foot, how high is the building?
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5.6 The Law of Sines
Let triangle ABC be any triangle with a, b, and c
representing the measures of the sides opposite
the angles with measurements A, B, and C
respectively. Then the following is true Ex
Solve triangle ABC if A 29o10, B 62o20,
and c 11.5. Round angle measures to the
nearest minute and side measures to the nearest
tenth.
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• When the measure of two sides of a triangle and
  the measure of the angle opposite one of them are
  given, there may not always be one solution.
  However one of the following will be true
• No triangle exists.
• Exactly one triangle exists.
• Two triangles exist.
• Case 1 angle A less than 90o
• If a b sin A, one solution exists
• If a lt b sin A, no solution exists
• If a gt b sin A, and a b one solution exists.
• If b sin A lt a lt b, two solutions exist.
• Case 2 angle A greater than 90o
• If a b, no solution exists.
• If a gt b, one solution exists.

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Ex Solve triangle ABC if A 63o10, b 18,
and a 17. Round angle measures to the nearest
minute and side measures to the nearest
tenth. Ex Solve triangle ABC if A 43o, b
20, and a 11. Round angle measures to the
nearest minute and side measures to the nearest
tenth.
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5.7 The Law of Cosines
Let triangle ABC be any triangle with a, b, and c
representing the measures of sides opposite
angles with measurements A, B, and C,
respectively. Then, the following are true
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Ex Solve triangle ABC if A 52o10, b 6, and
c 8. Round angle measures to the nearest
minute and side measures to the nearest
tenth. Ex Solve triangle ABC if a 21, b
16.7, and c 10.3. Round angle measures to the
nearest minute.
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5.8 Area of Triangles
We can create a new formula for area of any
triangle using our understanding of the law of
sines and cosines. Find the area of
triangle ABC if a 7.5, b 9, and C 100o.
Round your answer to the nearest tenth.
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Find the area of triangle ABC if a 18.6, A
19o20, and B 63o50. Round your answer to the
nearest tenth. Find the are of triangle ABC
if a , b 2, and c 3. Round your
answer to the nearest tenth.
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Herons Formula If the measures of the sides of a
triangle are a, b, and c, then the area, K, of
the triangle is found as follows Calculate
the area of triangle ABC if a 20, b 30, and c
40.
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If alpha is the measure of the central angle
expressed in radians and the radius of the circle
measures r units then the are of the segment S is
as follows A sector has a central angle of
150o in a circle with radius of 11.5 inches.
Find the area of the circular segment to the
nearest tenth.