Trigonometry
A brief review
1.4 Trigonometry
1.4 Trigonometry
ho
sin  
h
ha
cos  
h
ho
tan 
ha
1.4 Trigonometry
ho
tan 
ha
ho
tan 50 
67.2m

ho  tan50 67.2m  80.0m

1.4 Trigonometry
 ho 
  sin  
h
1
 ha 
  cos  
h
1
 ho
  tan 
 ha
1



1.4 Trigonometry
 ho
  tan 
 ha
1



 2.25m 

  tan 
  9.13
 14.0m 
1
1.4 Trigonometry
Pythagorean theorem:
h  h h
2
2
o
2
a
1.4.1. Which one of the following terms is not a trigonometric function?
a) cosine
b) tangent
c) sine
d) hypotenuse
e) arc tangent
1.4.1. Which one of the following terms is not a trigonometric function?
a) cosine
b) tangent
c) sine
d) hypotenuse
e) arc tangent
1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos ?
a) one
b) zero
c) sin1 
d) arc cos 
e) tan 
1.4.2. For a given angle , which one of the following is equal to the ratio of sin /cos ?
a) one
b) zero
c) sin1 
d) arc cos 
e) tan 
1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following
ratios is equal to the sine of the angle ?
a)
A
B
b)
A
C
c)
B
C
d)
B
A
e)
C
B
1.4.3. Referring to the triangle with sides labeled A, B, and C as shown, which of the following
ratios is equal to the sine of the angle ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following
ratios is equal to the tangent of the angle  ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.4. Referring to the triangle with sides labeled A, B, and C as shown, which of the following
ratios is equal to the tangent of the angle  ?
a)
b)
c)
d)
e)
A
B
A
C
B
C
B
A
C
B
1.4.5. Which law, postulate, or theorem states the following: “The square of the length of the
hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two
sides.”
a) Snell’s law
b) Pythagorean theorem
c) Square postulate
d) Newton’s first law
e) Triangle theorem
1.4.5. Which law, postulate, or theorem states the following: “The square of the length of the
hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two
sides.”
a) Snell’s law
b) Pythagorean theorem
c) Square postulate
d) Newton’s first law
e) Triangle theorem
What do you get when you marry measurement and trigonometry
VECTORS
1.5 Scalars and Vectors
A scalar quantity is one that can be described
by a single number:
temperature, speed, mass
A vector quantity deals inherently with both
magnitude and direction:
velocity, force, displacement
1.5 Scalars and Vectors
Arrows are used to represent vectors. The
direction of the arrow gives the direction of
the vector.
By convention, the length of a vector
arrow is proportional to the magnitude
of the vector.
4 lb
8 lb
1.5 Scalars and Vectors
1.5.1. Which one of the following statements is true concerning scalar quantities?
a) Scalar quantities have both magnitude and direction.
b) Scalar quantities must be represented by base units.
c) Scalar quantities can be added to vector quantities using rules of trigonometry.
d) Scalar quantities can be added to other scalar quantities using rules of ordinary addition.
e) Scalar quantities can be added to other scalar quantities using rules of trigonometry.
1.5.1. Which one of the following statements is true concerning scalar quantities?
a) Scalar quantities have both magnitude and direction.
b) Scalar quantities must be represented by base units.
c) Scalar quantities can be added to vector quantities using rules of trigonometry.
d) Scalar quantities can be added to other scalar quantities using rules of ordinary addition.
e) Scalar quantities can be added to other scalar quantities using rules of trigonometry.
1.5.2. Which one of the following quantities is a vector quantity?
a) the age of the pyramids in Egypt
b) the mass of a watermelon
c) the sun's pull on the earth
d) the number of people on board an airplane
e) the temperature of molten lava
1.5.2. Which one of the following quantities is a vector quantity?
a) the age of the pyramids in Egypt
b) the mass of a watermelon
c) the sun's pull on the earth
d) the number of people on board an airplane
e) the temperature of molten lava
1.5.3. A vector is represented by an arrow. What is the significance of the length of the arrow?
a) Long arrows represent velocities and short arrows represent forces.
b) The length of the arrow is proportional to the magnitude of the vector.
c) Short arrows represent accelerations and long arrows represent velocities.
d) The length of the arrow indicates its direction.
e) There is no significance to the length of the arrow.
1.5.3. A vector is represented by an arrow. What is the significance of the length of the arrow?
a) Long arrows represent velocities and short arrows represent forces.
b) The length of the arrow is proportional to the magnitude of the vector.
c) Short arrows represent accelerations and long arrows represent velocities.
d) The length of the arrow indicates its direction.
e) There is no significance to the length of the arrow.
1.5.4. Which one of the following situations involves a vector quantity?
a) The mass of the Martian soil probe was 250 kg.
b) The overnight low temperature in Toronto was 4.0 C.
c) The volume of the soft drink can is 0.360 liters.
d) The velocity of the rocket was 325 m/s, due east.
e) The light took approximately 500 s to travel from the sun to the earth.
1.5.4. Which one of the following situations involves a vector quantity?
a) The mass of the Martian soil probe was 250 kg.
b) The overnight low temperature in Toronto was 4.0 C.
c) The volume of the soft drink can is 0.360 liters.
d) The velocity of the rocket was 325 m/s, due east.
e) The light took approximately 500 s to travel from the sun to the earth.
1.6 Vector Addition and Subtraction
Often it is necessary to add one vector to another.
1.6 Vector Addition and Subtraction
3m
5m
8m
1.6 Vector Addition and Subtraction
1.6 Vector Addition and Subtraction
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
R  2.00 m  6.00 m
2
R
2
2
2.00 m  6.00 m
2
2
 6.32 m
R
2.00 m
6.00 m
1.6 Vector Addition and Subtraction
tan  2.00 6.00
  tan 2.00 6.00  18.4
1

6.32 m
2.00 m

6.00 m
1.6 Vector Addition and Subtraction
When a vector is multiplied
by -1, the magnitude of the
vector remains the same, but
the direction of the vector is
reversed.
1.6 Vector Addition and Subtraction

B
 
AB

A

A
 
AB

B
1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed due north.
Which of the following choices correctly indicates the directions of vectors A and B?
a) A is directed due west and B is directed due north
b) A is directed due west and B is directed due south
c) A is directed due east and B is directed due south
d) A is directed due east and B is directed due north
e) A is directed due north and B is directed due west
1.6.1. A and B are vectors. Vector A is directed due west and vector B is directed due north.
Which of the following choices correctly indicates the directions of vectors A and B?
a) A is directed due west and B is directed due north
b) A is directed due west and B is directed due south
c) A is directed due east and B is directed due south
d) A is directed due east and B is directed due north
e) A is directed due north and B is directed due west
1.7 The Components of a Vector


x and y are called the x vectorcomponent

and the y vectorcomponentof r.
1.7 The Components of a Vector

T hevectorcomponentsof A are two perpendicular


vectorsA x and A y thatare parallelto the x and y axes,
 

and add togethervectorially so thatA  A x  A y .
1.7 The Components of a Vector
It is often easier to work with the scalar components
rather than the vector components.
Ax and Ay are thescalar components

of A.
xˆ and yˆ are unit vectors with magnitude1.

A  Ax xˆ  Ay yˆ
1.7 The Components of a Vector
Example
A displacement vector has a magnitude of 175 m and points at
an angle of 50.0 degrees relative to the x axis. Find the x and y
components of this vector.
sin   y r




y  r sin   175m sin 50.0  134m
cos  x r
x  r cos  175m cos50.0  112m

r  112mxˆ  134myˆ
1.7.1. A, B, and, C are three vectors. Vectors B and C when added together
equal the vector A. In mathematical form, A = B + C. Which one of the
following statements concerning the components of vectors B and C must
be true if Ay = 0?
a) The y components of vectors B and C are both equal to zero.
b) The y components of vectors B and C when added together equal zero.
c) By  Cy = 0 or Cy  By = 0
d) Either answer a or answer b is correct, but never both.
e) Either answer a or answer b is correct. It is also possible that both are
correct.
1.7.1. A, B, and, C are three vectors. Vectors B and C when added together
equal the vector A. In mathematical form, A = B + C. Which one of the
following statements concerning the components of vectors B and C must
be true if Ay = 0?
a) The y components of vectors B and C are both equal to zero.
b) The y components of vectors B and C when added together equal zero.
c) By  Cy = 0 or Cy  By = 0
d) Either answer a or answer b is correct, but never both.
e) Either answer a or answer b is correct. It is also possible that both are
correct.
1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x axis. Which of
the following choices indicates the horizontal and vertical components of vector r?
a)
rx
+22 km/h
ry
+66 km/h
b)
+39 km/h
+79 km/h
c)
+79 km/h
+39 km/h
d)
+66 km/h
+22 km/h
e)
+72 km/h
+48 km/h
1.7.2. Vector r has a magnitude of 88 km/h and is directed at 25 relative to the x axis. Which of
the following choices indicates the horizontal and vertical components of vector r?
a)
rx
+22 km/h
ry
+66 km/h
b)
+39 km/h
+79 km/h
c)
+79 km/h
+39 km/h
d)
+66 km/h
+22 km/h
e)
+72 km/h
+48 km/h
1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A.
Vector A has a magnitude of 88 units and it is directed at an angle of 44 relative to the x axis as
shown. Find the scalar components of vectors B and C.
a)
Bx
63
By
0
Cx
0
Cy
61
b)
0
61
63
0
c)
63
0
61
0
d)
0
63
0
61
e)
61
0
63
0
1.7.3. A, B, and, C are three vectors. Vectors B and C when added together equal the vector A.
Vector A has a magnitude of 88 units and it is directed at an angle of 44 relative to the x axis as
shown. Find the scalar components of vectors B and C.
a)
Bx
63
By
0
Cx
0
Cy
61
b)
0
61
63
0
c)
63
0
61
0
d)
0
63
0
61
e)
61
0
63
0
1.8 Addition of Vectors by Means of Components
  
C AB

A  Ax xˆ  Ay yˆ

B  Bx xˆ  By yˆ
1.8 Addition of Vectors by Means of Components

C  Ax xˆ  Ay yˆ  Bx xˆ  B y yˆ
  Ax  Bx xˆ  Ay  B y yˆ
Cx  Ax  Bx
Cy  Ay  By
1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has scalar
components Bx = 22 m/s and By = 18 m/s. Determine the scalar components of vector C = A
 B.
a)
Cx
13 m/s
Cy
3 m/s
b)
57 m/s
33 m/s
c)
13 m/s
33 m/s
d)
57 m/s
3 m/s
e)
57 m/s 3 m/s
1.8.1. Vector A has scalar components Ax = 35 m/s and Ay = 15 m/s. Vector B has scalar
components Bx = 22 m/s and By = 18 m/s. Determine the scalar components of vector C = A
 B.
a)
Cx
13 m/s
Cy
3 m/s
b)
57 m/s
33 m/s
c)
13 m/s
33 m/s
d)
57 m/s
3 m/s
e)
57 m/s 3 m/s