vectors right triangle trigonometry. 9-1 the tangent ratio the ratio of the length to the opposite...
DESCRIPTION
Writing the Tangent The tangent of angle A is written as tanA =TRANSCRIPT
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Vectors
Right Triangle Trigonometry
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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
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Writing the Tangent
The tangent of angle A is written as
tanA = adjacentopposite
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Identifying Tangents
tanA =
tanB =
A
B
C1212
513
125
512
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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
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9-2 Sine and Cosine Ratios
Leg opposite angle A
Leg adjacent to angle A
Hypotenuse
Angle A
hypotenuseoppositeA sin
hypotenuseadjacentA cos
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Sine and Cosine
15
817
A
B
C
178sin A
1715cos A
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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
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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
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9-3 Angles of Elevation & Depression
Angle of Elevation- measured from the horizon up
Angle of Depression- measured from the horizon down
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Angle of elevation
x
The angle of elevation is the angle formed by the line of sight and the
horizontal
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Angle of depression
x
The angle of depression is the angle formed by the line of sight
and the horizontal
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Combining the two
x
x
elevationdepression
It’s alternate interior
angles all over again!
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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
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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
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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
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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
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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,
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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
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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
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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)
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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
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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
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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