Transcript Document

Vectors
Right Triangle Trigonometry
9-1 The Tangent Ratio
 The ratio of the length to the opposite
leg and the adjacent leg is the
B
Tangent of angle A
Leg
opposite
angle A
Angle A
A
C
Leg adjacent to
angle A
Writing the Tangent
 The tangent of angle A is written as
 tanA =
opposite
adjacent
Identifying Tangents
5
tanA =
12
B
12
tanB =
5
13
A
12
5
C
Tangent Inverse
 The Tangent Inverse allows you to
find the angle given the opposite and
adjacent sides from this angle.
 X=Tan-1(2/5)
x  21.8
0
2
x
5
9-2 Sine and Cosine Ratios
opposite
sin A 
hypotenuse
adjacent
cos A 
hypotenuse
Leg
opposite
angle A
Angle A
Leg adjacent to
angle A
Sine and Cosine
8
sin A 
17
B
17
15
cos A 
17
8
A
C
15
Sin-1 and Cos-1
 Angle A = sin-1(8/17)
Angle _ A  28.07
 Angle B =
0
B
cos-1(15/17)
Angle _ A  28.07
17
0
A
15
8
C
Keeping It Together
 Use the following acronym to help
you remember the ratios
 SOHCAHTOA
 Sine is Opposite over Hypotenuse
 Cosine is Adjacent over Hypotenuse
 Tangent is Opposite over Adjacent
9-3 Angles of Elevation &
Depression
 Angle of Elevation- measured from
the horizon up
 Angle of Depression- measured from
the horizon down
Angle of elevation
The angle of
elevation is the
angle formed by
the line of sight
and the
horizontal
x
Angle of depression
x
The angle of
depression is the
angle formed by
the line of sight
and the horizontal
Combining the two
x
depression
It’s alternate
interior angles
all over again!
elevation
x
The angle of elevation of building A to building B is 250. The
distance between the buildings is 21 meters. Calculate how much
taller Building B is than building A.
Step 1: Draw a right
angled triangle with the
given information.
Step 2: Take care with
placement of the angle of
elevation
Step 3: Set up the trig
equation.
Step 4: Solve the trig
equation.
Angle of
elevation
B
hm
A
250
21
tan 25 
h
21
h  21 tan 25
h  9.8 m (1 dec. pl )
A boat is 60 meters out to sea. Madge is standing on a cliff 80
meters high. What is the angle of depression from the top of the
cliff to the boat?
Step 1: Draw a right angled
triangle with the given
information.
Step 2: Use your knowledge of
alternate angles to place  inside
the triangle.
Step 3: Decide which trig
ratio to use.
Step 4: Use calculator to find
the value of the unknown.

Angle of
depression
80 m

60 m
80
tan  
60
1  80 
  tan  
 60 
  53.1o
9-4 Vectors
 Vector- a quantity with magnitude
(the size or length) and direction, it is
represented by an arrow
 Initial Point- is where the vector
starts, i.e., the tail of the arrow
 Terminal Point- is where the
arrow stops, i.e., the point of the
arrow
Vectors
 The magnitude corresponds to the
distance from the initial point to the
terminal point. The symbol for the
magnitude of a vector is V .
 The symbol for a vector is an arrow
over a lower case letter, a or capital
letters of the initial and terminal
points
 The distance corresponds to the
direction in which the arrow points
Describing Vectors
 An ordered pair in a coordinate plane can
also be used for a vector.
 The magnitude is the cosine and the
direction is the sine. The ordered pair is
written this way, x, y , to indicate a
vectors distance from the origin.
 A vector with the initial point at the origin
is said to be in Standard Position.
Describing Vectors in the
Coordinate Plane
 With a vector in Standard Position,
the coordinates of the terminal point
describes the vector.
 The magnitude is the hypotenuse of a
right triangle. The cosine of the
direction angle is the x coordinate
and the sine is the y coordinate
 See Example 1 on Pg. 490
Describing a Vector Direction
 Vector direction commonly uses
compass directions to describe a
vector.
 The direction is given as a number of
degrees east, west, north or south of
another compass direction, such as
250 east of north
 See Example 2 Pg. 491
Vector Addition
 A vector sum is called the
RESULTANT.
 Adding vectors gives the result of
vectors that occur in a sequence (See
the top of pg. 492) or that act at the
same time (See Examples 4 & 5 pgs.
492, 493)
9-5 Trig Ratios and Area
 Parts of Regular Polygons
 Center- a point equidistant from the
vertices
 Radius- a segment from the center to a
vertex
 Apothem- a segment from the center
perpendicular to a side
 Central Angle- angle formed by two radii
Finding Area in a Regular Polygon
 Formula for Area
 A=(apothem X perimeter) divided by 2
 Use the trig ratio, and the central
angle to find the apothem or a side
for the perimeter.
 See Examples 1 & 2 pgs. 498-499
Area of a Triangle Given SAS
 Theorem 9-1
 The area of a triangle is one half the
product of the lengths of the sides and
the sine of the included angle.
bc(sin A)
A
2
Where b and c are sides and A is the angle
between them. See the bottom of pg 499
and Example 3 pg. 500