Download 2.1 Definition II

2.1 Definition II: Right Angle Trigonometry (1 of 26)
2.1 Definition II: Right Angle Trigonometry
The Greek translation of trigonometry is “triangle measure.” We
now define the trigonometric functions as ratios of the side lengths
of right triangles.
Definition II: Right Angle Trigonometry
If triangle ABC is a right triangle with C = 90 0 . Then the six
trigonometric functions for angle A are defined as follows:
Example 1
Triangle ABC is a right triangle with C = 90 0 . If
a = 6 and c = 10 , find the six trigonometric functions of A.
2.1 Definition II: Right Angle Trigonometry (2 of 26)
Example 2
Use Definition II to explain why it is impossible
to have sin θ = 2 .
Definition of Cofunctions
Sine and cosine are cofuncitons, as are tangent and cotangent, and
secant and cosecant. We say sine is the cofunction of cosine and
cosine is the cofunciton of sine.
Find the Six Trig Functions for Angle B
Triangle ABC is a right triangle with C = 90 0 . Find the six
trigonometric functions for angle B.
2.1 Definition II: Right Angle Trigonometry (3 of 26)
Fact: If A + B = 90 0 ; that is, A and B are complements of each
other, then every trigonometric function of angle A equals
its cofunction of angle B.
Cofunction Theorem
A trigonometric function of an angle θ is equal to the cofunction
of the complement 90 0 − θ .
Example 3
Fill in the blanks so each statement is true.
a.
cos 30 0 = sin____
b.
tan y = cot____
c.
sec 75 0 = tan____
2.1 Definition II: Right Angle Trigonometry (4 of 26)
Fill in the following table.
Table 1: Exact Values (Memorize)
θ
sin θ
cos θ
tan θ
30 0
45 0
60 0
Example 4
Use exact values to verify each statement is true.
a.
cos 2 30 0 + sin 2 30 0 = 1
b.
cos 2 45 0 + sin 2 45 0 = 1
2.1 Definition II: Right Angle Trigonometry (5 of 26)
Example 5
Let x = 30 0 and y = 45 0 . Simplify each of the
following as much as possible.
a.
2 sin x
b.
sin 2y
c.
4 sin(3x − 90 0 )
Example 5b
Fill in the following table.
Table 1: Exact Values (Memorize)
θ
sin θ
cos θ
00
90 0
tan θ
2.1 Definition II: Right Angle Trigonometry (6 of 26)
Trigonometric Function Values at Common Angles
(Memorize)
2.2 Calculators and Trig Function of and Acute Angle (7 of 26)
2.2 Calculators and Trigonometric Functions of an
Acute Angle
Degrees, Minutes, Seconds
Since there are 360 0 in one complete revolution, 10 = 1 / 360
revolution. If we divide one degree into 60 equal parts, each part is
called one minute, denoted 1′ . So, there are 60 degrees in one
minute, and one minute is 1/60 degree. The next smaller unit of
angle measurement is a second. Sixty seconds equals one minute,
denoted 60 ′′ = 1′ and 1′′ = (1 / 60 )′ .
10 = 60 ′
or
1′ = ( 601 )
1′ = 60 ′′
or
1′′ = ( 601 )′
0
DMS Notation
Example 1
Add 48 0 49 ′ + 72 0 26 ′
2.2 Calculators and Trig Function of and Acute Angle (8 of 26)
Example 2
Subtract 24 014 ′12 ′′ from 90 0
Example 3
Change 27.25 0 to degrees and minutes.
Example 4
Subtract 10 012 ′ to decimal degrees.
Example 5
Use a calculator to find cos 37.8 0
Example 6
Use a calculator to find tan 58.75 0
Example 7
Use a calculator to find sin 2 14 0
Example 8
Use a calculator to find sec 78 0
Example 9
Use a calculator to verify
sin 24.30 = cos(90 0 − 24.30 )
2.2 Calculators and Trig Function of and Acute Angle (9 of 26)
Use a function and table features to estimate the values to 4
decimal places.
x
0
30
45
sin x
0
1
2 = 0.5000
1
≈ 0.7071
2
60
3
2
90
cos x
tan x
≈ 0.8660
1
Inverse Trigonometric Functions
The sin, cos and tan keys allow us allow us to find the value a
trigonometric function (that is, one of the six ratios) when we input
an angle. When we are given a trigonometric ratio the inverse
trigonometric functions sin −1 , cos −1 and tan −1 give us the
corresponding angle.
Example 10
Find the acute angle θ for which tan θ = 3.152 .
Round to the nearest tenth of a degree.
2.2 Calculators and Trig Function of and Acute Angle (10 of 26)
Example 11
Find the acute angle A for which sin A = 0.3733 .
Round to the nearest tenth of a degree.
Example 12
Find the acute angle B for which sec B = 1.0768 .
Round to the nearest hundredth of a degree.
Example 13
Find the acute angle C for which cot C = 0.0975 .
Round to the nearest degree.
2.3 Solving Right Triangles (11 of 26)
2.3 Solving Right Triangles
Significant Digits
The number of significant digits in a number is found by counting
all the digits from left to right beginning with the first nonzero
digit on the left. When no decimal point is present, the trailing
zeros are not considered significant.
How many significant digits are in each number?
a.
0.079
b.
0.0006
c.
54.2
d.
6.000
e.
9,200.
f.
2000
Significant Digits and Accuracy of Angle
2.3 Solving Right Triangles (12 of 26)
Example 1
In the right triangle ABC, A = 40 and c = 12
centimeters. Find a, b and B.
Example 2
In the right triangle ABC, a = 2.73 and
b = 3.41 centimeters. Find the remaining side and
angles.
2.3 Solving Right Triangles (13 of 26)
Example 3
The circle in Figure 3 has its center at C and a
radius of 18 inches. If triangle ADC is a right
triangle and A = 35 , find x, the distance from A
to B.
Example 4
Use the information Figure 4 to solve for x, the
distance from D to C.
2.3 Solving Right Triangles (14 of 26)
Example 5
Figure 5 is a simplified model of a Ferris wheel
whose diameter is 250 feet and whose height is 264 feet. If θ is the
central angle formed as a rider moves from P0 to P1 , find r, the
riders height above the ground when θ = 45 .
2.5 Vectors: A Geometric Approach (15 of 26)
2.4 Applications
Example 1
The two equal sides of an isosceles triangle are
each 24 centimeters. If each of the two equal angles measures 52 ,
find the length of the base and altitude.
Angle of Elevation and Angle of Depression
2.5 Vectors: A Geometric Approach (16 of 26)
Example 2
If a 75 foot flagpole casts a shadow 43 feet long,
to the nearest 10 minutes what is the angle of elevation of the sun
from the tip of the shadow?
Example 3
A man climbs 213 meters up the side of a pyramid
and finds that the angle of depression to his starting point is 52.6 .
How high off the ground is he?
2.5 Vectors: A Geometric Approach (17 of 26)
Example 4
Suppose Stacy and Amy are climbing Bishop’s
Peak. Stacey is at position S, and Amy is at position A. Find the
angle of elevation from Stacey to Amy.
2.5 Vectors: A Geometric Approach (18 of 26)
Bearings
Example 5
San Luis Obispo, California, is 12 miles due north
of Grover Beach. If Arroyo Grande is 4.6 miles due east of Grover
Beach, what is the bearing of San Luis Obispo from Arroyo
Grande?
Example 6
A boat travels on a course of bearing N 52 40 ′ E
for a distance of 238 miles. How may miles north and how many
miles each has the boat traveled?
2.5 Vectors: A Geometric Approach (19 of 26)
Example 7
Figure 10 is a diagram that shows how Diane
estimates the height of a flagpole. She can’t measure the distance
between herself and the flagpole directly because there is a fence
in the way. So she stands at point A facing the pole and finds the
angle of elevation from point A to the top of the pole to be 61.7 .
The she turns 90 and walks 25.0 feet to point B, where she
measures the angle between her path and a line form B to the base
of the pole. She finds the angle is 54.5 . Find the height of the
pole.
2.5 Vectors: A Geometric Approach (20 of 26)
Example 8
A helicopter is hovering over the desert when it
develops mechanical problems and is forced to land. After landing,
the pilot radios his position to a pair of radar stations located 25
miles apart along a straight road running north and south. The
bearing of the helicopter from one station is N 13 E, and from the
other it is S 19 E. After some computation one of the stations
instructs the pilot to walk die west for 3.5 miles to reach the road.
Is this information correct?
2.5 Vectors: A Geometric Approach (21 of 26)
2.5 Vectors: A Geometric Approach
A vector is a quantity with both a magnitude and direction.
Quantities with only a magnitude (“size”) are called scalars. Some
examples of vector quantities are gravity, velocity, force and work.
For example, a car traveling north at 50 mph has a different vector
than a car traveling south at 50 mph. Vectors can be represented as
an arrow, where the length of the arrow represents the vector’s
magnitude and the direction of the arrow is the direction of the
vector. Some velocity examples of vectors are:
Vector Notation
To distinguish between vectors and scalars, your textbook
represents vectors in bold face type, such as, U and V. However,

when you write them on paper, put an arrow above them (i.e. U
and V ). The absolute value symbol denotes the magnitude of a

vector. That is the magnitude of V is written V .
2.5 Vectors: A Geometric Approach (22 of 26)
The Zero Vector and Equivalent Vectors
The vector whose magnitude is zero is called
the zero vector, denoted 0. The zero vector
has no defined direction. Two vectors are
equivalent if they have the same magnitude
and direction. Position of a vector in space is
not important.
The sum of two vectors U and V, written U + V, is called the
resultant vector. It is found by placing the vectors tail to end or
using the parallelogram method as illustrated here:
To subtract two vectors, simply add the opposite. That is,
U - V = U + (-V)
2.5 Vectors: A Geometric Approach (23 of 26)
Example 1
A boat is crossing a river that runs due
north. The boat is pointed due east and is moving through the
water at 12 mph. If the current of the river is 5.1 mph, find the
actual course of the boat through the water to two significant
digits.
Horizontal and Vertical Vector Components
Notice the actual course of the boat is the
sum of the boat’s east (horizontal) vector
and the river’s north (vertical) vector.
In general to write a vector as the sum of its
horizontal and vertical vectors we must
superimpose a coordinate system write the
vector with its starting point at the origin,
this is called standard position. The vector
V shown has magnitude 15 making an angle
of 52 with the horizontal.
The vector V = Vx + Vy is shown as the sum
of its horizontal and vertical vectors.
Find Vx and Vy
2.5 Vectors: A Geometric Approach (24 of 26)
Example 2
A human cannonball is shot from a cannon
with an initial velocity of 53 mph at an angle of 60 from the
horizontal. Find the magnitudes of the horizontal and vertical
components of the initial velocity vector.
Example 3
An arrow is shot into the air so that its
horizontal velocity is 25 feet per second and its vertical velocity is
15 feet per second. Find the velocity V of the arrow.
2.5 Vectors: A Geometric Approach (25 of 26)
Example 4
A boat travels 72 miles bearing N 27 E and
then changes course to travel 37 miles at N 55 E. How far north
and how far east has the boat traveled on this 109 mile trip?
Force and Static Equilibrium
Suppose a 10 lb statue is setting on a table. The
force due to gravity is 10 lb downward (toward the
center of the earth). The statue does not move
because the table exerts an equal and opposite
force of 10 lb upward. The sum of the two force
vectors acting on the statue is 0. When an object is
stationary we say it is in a state of static
equilibrium.
2.5 Vectors: A Geometric Approach (26 of 26)
Example 5
Danny is 5 years old and
weighs 420 pounds. He is sitting on a swing and
his sister Stacey pulls him and the swing back
horizontally through an angle of 30.0 and stops.
Find the tension in the ropes of the swing and the
magnitude of the force exerted by Stacey.
Work
Work is a measure of effort used when moving an object while
applying a force to it. If a constant force F is applied to an object
and moves the object in a straight line a distance d in the direction
of the force, then the work W performed by the force is W = F d .
Example 6
A shipping clerk
pushes a heavy package across the floor.
He applies a force of 64 pounds in a
downward direction making an angle of
35 with the horizontal. If the package is
moved 25 feet, how much work is done
by the clerk?