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