Unit 4.2 Right Triangles/ Vectors

1

Unit 4.2

Right Triangles/

Vectors

The trigonometric functions of a right triangle, with

an angle θ, are defined by ratios of two sides of the

triangle.

The sides of the right triangle are:

OPP the side opposite the angle θ

ADJ the side adjacent to the angle θ

HYP is the hypotenuse of the right triangle.

opp

adj

hyp

θ

SOH – CAH - TOA

sine, cosine, tangent

opp

adj

hyp

Trigonometric

Functions

sin θ = cos θ = tan θ =

hyp

adj

adj

opp

hyp

opp

θ

Example #1:

a. What is sine, cosine and tangent for Angle Y?

b. Using sine, what is the value of Angle Y? (Use sin-1 on your

calculator)

c. Using cosine, what is the value for Angle Y?

d. What is the value for angle X?

10.5

22.8

25.1

X

Y

Example #2:

a. What is the length of Side X?

b. What is the length of Side Y?

20.1

Side X

Side Y

61.1º

Example #3

All of the triangles in the previous questions follow an equation.

Note the following symbols:

c = hypotenuse

a = any side of the triangle other than the hypotenuse

b = any side of the triangle other than the hypotenuse

Which of the following equations is true for all right triangles?

a. a + b = c

b. a2 + b2 = c2

c. a2 - b2 = c2

d. all of these

Example #4

a. What is the length of the missing side?

18.7

X

42.9

Example #5

a. What is the length of the missing side?

12.9

23.1

X

A surveyor is standing 115 feet from the base of the

Washington Monument. The surveyor measures the angle of

elevation to the top of the monument as 78.3. How tall is

the Washington Monument?

Example #6

Solution:

Where adj = 115 and opp (x) is the height of the monument. So, the

height of the Washington Monument is

tan(78.3) = x/ 115

X = 115(4.82882) 555 feet. 78.3°

115 feet

X

Vectors /

Parallelogram Method

Scalar: A quantity with magnitude only.

Vector: A quantity with magnitude & direction.

A diagram or sketch is helpful & vital!

I don’t see how it is possible to solve a vector problem

without a diagram!

Vectors

1. In order to show direction and speed of an object,

vectors are used.

2. A vector is a mathematical quantity that has both

a magnitude (length) and direction.

3. A vector has an initial point (head), and a terminal

point (tail).

Vectors

Velocity being a vector quantity

Example: 1.3 m/s @ 20° N of W

1.3 m/s: Magnitude (Length of vector)

20° N of W: Direction (Direction the vector points)

Velocity is a vector quantity: Direction

Drawing a Vector:

Initial Point

(Head)

P

Terminal Point

(Tail)

Q

It does not matter where a vector is located in a

plane, as long as it maintains the same direction

and magnitude.

For example, all the vectors below are equal.

Example:

Airplane traveling 50 m/s E

Graphically

VP = 50 m/s east

Adding Vectors /

(2-Vector Situations)

Collinear (Same or opposite directions)

Two velocities acting in the same direction;

add magnitudes and keep the direction.

Example:

Airplane with a tailwind or Boat traveling downstream

Mathematically

VR = VB + VW

VR = 50 m/s downstream + 40 m/s downstream

VR = 90 m/s downstream

Graphically

VB = 50 m/s down VW = 40 m/s down VR = 90 m/s downstream

Two velocities acting in opposite directions; Example: Airplane with a head wind or boat traveling upstream Mathematically VR = VP + VW

VR = 50 m/s E + 40 m/s W VR = 50 m/s E + (-40 m/s E) VR = 10 m/s E Graphically

VP = 50 m/s east VW = 40 m/s west VR = 10 m/s east

What is the ground speed of an airplane

flying with an air speed of 100 mph into a

headwind of 100 mph?

Solve this problem using vectors

Adding Vectors /

(2-Vector Situations)

Perpendicular (90°)

Vector Addition: (90 Degrees)

Mathematically: Trigonometry (sin, cos, tan)

Graphically: Parallelogram Method

i. When adding two vectors that share the same tail,

There is one origin point for both vectors.

ii. We will use this method for two vectors only!!!!

iii. Draw the first vector again by placing its tail on the

head of the second vector. Then draw the second

vector by placing its tail on the head of the first vector.

The diagonal is the resultant vector. HUHH?

Parallelogram Method

+

The Black Vector represents the RESULTANT VECTOR

(VR) of the red and gray vectors.

VY Vx

Vx

VY

Vx

VY

Vx

VY

θ

VY

Vx

Math Examples:

Example #1:

A plane flies 30 m/s directly south and a 60 m/s wind is

blowing east. Find the magnitude and direction of the

planes resultant velocity.