Vectors Right Triangle Trigonometry. 9-1 The Tangent Ratio The ratio of the length to the opposite...
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Transcript of Vectors Right Triangle Trigonometry. 9-1 The Tangent Ratio The ratio of the length to the opposite...
Vectors
Right Triangle Trigonometry
9-1 The Tangent Ratio The ratio of the length to the opposite
leg and the adjacent leg is the Tangent of angle A
A C
B
Angle A
Leg opposite angle A
Leg adjacent to angle A
Writing the Tangent
The tangent of angle A is written as
tanA = adjacentopposite
Identifying Tangents
tanA =
tanB =
A
B
C1212
513
125
512
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
2
5
08.21x
9-2 Sine and Cosine Ratios
Leg opposite angle A
Leg adjacent to angle A
Hypotenuse
Angle A
hypotenuseoppositeA sin
hypotenuseadjacentA cos
Sine and Cosine
15
817
A
B
C
178sin A
1715cos A
Sin-1 and Cos-1
Angle A = sin-1(8/17)
Angle B = cos-1(15/17)
AC
B
15
178
007.28_ AAngle
007.28_ AAngle
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
x
The angle of elevation is the angle formed by the line of sight and the
horizontal
Angle of depression
x
The angle of depression is the angle formed by the line of sight
and the horizontal
Combining the two
x
x
elevationdepression
It’s alternate interior
angles all over again!
B
A21
h m
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 3: Set up the trig equation.
).1(8.9
25tan21
pldecmh
h
Angle of elevation
Step 4: Solve the trig equation.
2125tan h
250
Step 2: Take care with placement of the angle of elevation
Step 1: Draw a right angled triangle with the given information.
Step 3: Decide which trig ratio to use.
60 m
80 m
6080tan
Step 4: Use calculator to find the value of the unknown. o1.53
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 2: Use your knowledge of alternate angles to place inside the triangle.
Angle of depression
6080tan 1
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 .
The symbol for a vector is an arrow over a lower case letter, or capital letters of the initial and terminal points
The distance corresponds to the direction in which the arrow points
V
a
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, , to indicate a vectors distance from the origin.
A vector with the initial point at the origin is said to be in Standard Position.
yx,
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.
Where b and c are sides and A is the angle between them. See the bottom of pg 499 and Example 3 pg. 500
2)(sin AbcA