MATH 151 Engineering Math I, Spring 2014
JD Kim
Week1 Appendix D, Section 1.1 Introduction, Trigonometry review, Two-dimensional
vectors
Domain of a function
Ex1) Find the domain of f (x) =
Ex2) Find the domain of f (x) =
Ex3) Find the domain of f (x) =
x+1
x−2
x2
x+4
− 2x − 3
√
√
8−x+
1
x2 − 4
Ex4) If f (x) =
1
and g(x) = tan x, find
x−1
(4-1) f (g(x))
(4-2) g(f (x))
(4-3)
f (x + h) − f (x)
h
Measurement of Angles
Angle can be measured in degrees or radians. The angle given by a complete revolution contains 360◦ , or 2π radians
Ex5) Find radian measure of 60◦ .
2
Ex6) Express
5π
rad in degrees.
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The Trigonometric Functions
SOH CAH TOA
3
2
3π
Ex7) If cos θ = and
< θ < 2π, find the other five trigonometric functions
5
2
of θ.
Trigonometric Identities
is a relationship among the trigonometric functions.
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
1 + cot2 θ = csc2 θ
sin(−θ) = − sin θ
cos(−θ) = cos θ
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sin(θ + 2π) = sin θ
cos(θ + 2π) = cos θ
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos2 θ − sin2 θ
Ex8) Solve the following equations for x, 0 ≤ x ≤ 2π.
(8-1) 2 cos2 x − 1 = 0
(8-2) 2 cos x + sin 2x = 0
5 3π
< x < 2π, find sin(2x).
Ex9) If sec x = ,
3 2
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Graphs of the Trigonometric Functions
Plot points for 0 ≤ θ ≤ 2π and then using the periodic nature of the function to
complete the graph.
6
Section 1.1 Vectors
Definition: A Vector is a quantity that has both magnitude and direction.
A two-dimensional vector is an ordered pair ~a =< a1 , a2 > of real numbers.
The numbers a1 and a2 are called the components of ~a. a1 is x component,
a1 is y component.
Vector has an initial point and terminal point
If the initial point is origin, we call the position vector
~ is < x2 − x1 , y2 − y1 >
Given the points A(x1 , y1 ) and B(x2 , y2 ), the vector AB
Ex10) Find the vector presented by the directed line segment with initial point
A(1, 2) and terminal point B(4, −3).
The length of the vector ~a =< a1 , a2 > is |a| =
p
a21 + a22
Vector Addition
If ~a =< a1 , a2 > and ~b =< b1 , b2 >, then the vector ~a + ~b is defined by
~a + ~b =< a1 + b1 , a2 + b2 >
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Ex11) Find the component of the vector ~r given that;
(11-1) |~r| = 2 and ~r makes an angle of 60◦ with the positive x-axis.
(11-2) |~r| = 7 and ~r makes an angle of 150◦ with the positive x-axis.
(11-3) |~r| =
1
and ~r makes an angle of −45◦ with the positive x-axis.
2
Unit Vector is a vector with length one,
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~a
.
|~a|
Basis Vector (Standard) ~i =< 1, 0 >,
~j =< 0, 1 >
Ex12) If ~a =< 4, 3 > and ~b =< −2, 1 >, find |a| and the vectors a + b, a − b, 3b,
2a + 5b, a unit vector of b, and a vector with length 3 in the direction of b.
Ex13) Suppose a =< 1, 5 >, b =< 3, −1 > and c =< 8, 6 >, find scalars t and w
so that ta + wb = c.
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Applications to Physics and Engineering
A force is represented by a vector because it has both a magnitude and direction.
If several forces are acting on an object, Resultant Force experienced by the object
is the vector sum of the forces.
Ex14) John walks due west on the deck of a ship at 3mph. The ship is moving
north at 22mph. Find the speed and direction of John relative to the surface of the
water.
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~ and T~ , are acting on an object at a point P. |S|
~ = 20 pounds
Ex15) Two forces S
and measures a reference angle of 45◦ , |T~ | = 16 pounds and measures a reference
angle of 30◦ . Find the resultant force as well as its magnitude and direction.
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Ex16) Suppose that a wind is blowing from the direction N45◦ W at a speed of
50 km/h. A pilot is steering a plane in the direction N60◦ E at an speed of 250 km/h.
Find the true course (direction of the resultant velocity vector of the plane and wind)
and ground speed(magnitude of resultant).
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Ex17) A 100 lb weight hangs from two wires as shown in Figure. Find the tensions
(forces) T1 and T2 in both wires and their magnitudes.
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