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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.