Transcript Chapter 5 – The Trigonometric Functions

Chapter 5 – The Trigonometric
Functions
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
Change 30o to radians
Change
3 radians to degrees.
4
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
11
coterminal with 4 .
EX: Identify all angles that are coterminal with a 60o angle.
Reference Angle Rule
For any angle  , 0    2 , its reference angle
  is defined by
a.  when the terminal side is quadrant I
b.  when the terminal side is quadrant II
c.  when the terminal side is in quadrant III
d.2   when the terminal side is in quadrant IV
Ex: Find the measure of the reference angle for each.
5
4

510o
13
3
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.
s  r
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

vr
t
Where

is the angular velocity in radians per unit of time.
t
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 truck’s tires have a diameter of 26 inches. What is
the angular velocity of the wheels in revolutions per second?
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:
1 2
A r 
2
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.
5.3 Circular Functions
Def: If the terminal side of an angle
in standard position intersects
the unit circle at P(x,y), then cos   x andsin   y.
Find each value:
sin 90
o
cos 
For any angle in standard position with measure , a point P(x,y)
on its terminal side, and r  x 2  y 2 , the sine and cosine functions
ofare as follows:
sin  
y
r
cos  
x
r
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.
5
Find sin  when cos   and the terminal side of  is in the first
13
quadrant.
For any angle in standard position with measure , a point P(x,y)
on its terminal side, and r  x 2  y 2 , the trigonometric functions
ofare as follows:
sin  
y
r
cos  
x
r
csc  
r
y
sec  
r
x
tan  
csc  
y
x
x
y
The terminal side of an anglein standard position contains the
point with coordinates (8, -15). Find tangent, cotangent, secant,
and cosecant for .
If csc   2 andlies in quadrant III, find sine, cosine, tangent,
cotangent, and secant for .
5.4 Trigonometric Functions of Special
Angles

3
Let’s first discuss the value of each of the trig functions at ,  , , 2
2
2
Then let’s create two very special triangles which we will then use to
help find many other trig values.
5.5 Right Triangles
For an acute angle A in a right triangle ABC, the trigonometric
functions are as follows:
Opposite
a

Hypotenuse c
Adjacent
b
cos A 

Hypotenuse c
Opposite a
tan A 

Adjacent b
sin A 
B
c
A
Hypotenuse c

Opposite
a
Hypotenuse c
sec A 

Adjacent
b
Adjacent b
cot A 

Opposite a
csc A 
a
b
C
For the following triangle find the values of the six trig functions of A.
13
5
A
12
Solve right triangle ABC. Round angle measures to the nearest
degree and side measures to the nearest tenth. A = 49o
A
c
b
C
7
B
In the given triangle find the measure of angle R to the nearest
T
degree.
14
8
S
R
Assume that a ladder is mounted 8ft off the ground.
a. How far from an 84ft burning building should the base of the
ladder be placed to achieve the optimum operating angle of
60o?
b. How far should the ladder be extended to reach the roof?
A flagpole 40ft high stands on top of the Wentworth Building.
From a point P in front of Bailey’s 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?
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:
a
b
c


sin A sin B sin C
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.
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:
1. No triangle exists.
2. Exactly one triangle exists.
3. Two triangles exist.
Case 1: angle A less than 90o
a. If a = b sin A, one solution exists
b. If a < b sin A, no solution exists
c. If a > b sin A, and a 
b one solution exists.
d. If b sin A < a < b, two solutions exist.
Case 2: angle A greater than 90o
a. If a 
b, no solution exists.
b. If a > b, one solution exists.
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.
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:
a 2  b 2  c 2  2bc cos A
b2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
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.
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.
1
1
1
K  bc sin A  ac sin B  ab sin C
2
2
2
1 2 sin A sin B 1 2 sin A sin B 1 2 sin A sin B
K c
 a
 b
2
sin C
2
sin C
2
sin C
Find the area of triangle ABC if a = 7.5, b = 9, and C = 100o.
Round your answer to the nearest tenth.
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 =
answer to the nearest tenth.
2
, b = 2, and c = 3. Round your
Heron’s 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:
1
K  s  s  a  s  b  s  c  ; s   a  b  c 
2
Calculate the area of triangle ABC if a = 20, b = 30, and c = 40.
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:
S
1 2
r   sin  
2
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.