Transcript Chapter 7 Section 1

8.1 The Law of Sines
Congruency and Oblique Triangles ▪ Derivation of the Law of
Sines ▪ Using the Law of Sines ▪ Ambiguous Case ▪ Area of a
Triangle
Congruence Axioms
Side-Angle-Side (
)
If two sides and the included angle of one triangle are equal,
respectively, to two sides and the included angle of a second
triangle, then the triangles are congruent.
Angle-Side-Angle (
)
If two angles and the included side of one triangle are equal,
respectively, to two angles and the included side of a second
triangle, then the triangles are congruent.
Side-Side-Side (
)
If three sides of one triangle are equal, respectively, to three sides of
a second triangle, then the triangles are congruent.
Oblique Triangles
Oblique triangle: A triangle that is not a ____________ triangle
The measures of the three sides and the three angles of a triangle can be
found if at least one side and any other two measures are known.
Data Required for Solving Oblique Triangles
Case 1
Case 2
One side and two angles are known (SAA or ASA).
Two sides and one angle not included between the two
sides are known (SSA). This case may lead to more than
one triangle.
Case 3
Two sides and the angle included between the
two sides are known (SAS).
Case 4
Three sides are known (SSS).
Note - If three angles of a triangle are known, unique side lengths
cannot be found because AAA assures only similarity, not congruence.
Derivation of the Law of Sines
Let h be the length of the perpendicular from vertex B
to side AC (or its extension) of an oblique triangle.
Then c is the hypotenuse of right triangle ABD, and
a is the hypotenuse of right triangle BDC.
In triangle ADB,
In triangle BDC,
Law of Sines
In any triangle ABC, with sides a, b, and c,
Example 1
USING THE LAW OF SINES TO SOLVE A
TRIANGLE (SAA)
Solve triangle ABC if A = 32.0°, C = 81.8°, and a = 42.9 cm.
Example 2
USING THE LAW OF SINES IN AN
APPLICATION (ASA)
Jerry wishes to measure the distance across the Big Muddy River. He
determines that C = 112.90°, A = 31.10°, and b = 347.6 ft. Find the distance
a across the river.
Example 3
USING THE LAW OF SINES IN AN
APPLICATION (ASA)
Two ranger stations are on an east-west line 110 mi
apart. A forest fire is located on a bearing N 42° E
from the western station at A and a bearing of N 15°
E from the eastern station at B. How far is the fire
from the western station?
Description of the Ambiguous
Case
If the lengths of two sides and the angle opposite one of them are given
(Case 2, SSA), then zero, one, or two such triangles may exist.
Applying the Law of Sines
1.
2.
3.
For any angle θ of a triangle,
0 < sin θ ≤ 1. If sin θ = 1, then θ = 90° and the triangle is a right
triangle.
sin θ = sin(180° – θ) (Supplementary angles have the same sine
value.)
The smallest angle is opposite the shortest side, the largest angle is
opposite the longest side, and the middle-value angle is opposite
the intermediate side (assuming the triangle has sides that are all of
different lengths).
Example 4
SOLVING THE AMBIGUOUS CASE (NO
SUCH TRIANGLE)
Solve triangle ABC if B = 55°40′, b = 8.94 m, and a = 25.1 m.
Note - In the ambiguous case, we are given two sides and
an angle opposite one of the sides (SSA).
Example 5
SOLVING THE AMBIGUOUS CASE (TWO
TRIANGLES)
Solve triangle ABC if A = 55.3°, a = 22.8 ft, and b = 24.9 ft.
Number of Triangles Satisfying the Ambiguous Case (SSA)
Let sides a and b and angle A be given in triangle ABC. (The
law of sines can be used to calculate the value of sin B.)
1.
If applying the law of sines results in an equation having
sin B > 1, then no triangle satisfies the given conditions.
2.
If sin B = 1, then one triangle satisfies the given
conditions and B = 90°.
3.
If 0 < sin B < 1, then either one or two triangles satisfy the
given conditions.
(a)
If sin B = k, then let B1 = sin–1 k and use B1 for B in
the first triangle.
(b) Let B2 = 180° – B1.
If A + B2 < 180°, then a second triangle exists. In
this case, use B2 for B in the second triangle.
Example 6
SOLVING THE AMBIGUOUS CASE (ONE
TRIANGLE)
Solve triangle ABC given A = 43.5°, a = 10.7 in., and c = 7.2 in.
Example 7
ANALYZING DATA INVOLVING AN
OBTUSE ANGLE
Without using the law of sines, explain why A = 104°, a = 26.8 m, and b =
31.3 m cannot be valid for a triangle ABC.
Area of a Triangle (SAS)
In any triangle ABC, the area A is given by the following formulas:
Example 8
FINDING THE AREA OF A TRIANGLE (SAS)
Find the area of triangle ABC.
Example 9
FINDING THE AREA OF A TRIANGLE
(ASA)
Find the area of triangle ABC if A = 24°40′, b = 27.3 cm, and C = 52°40′.
8.2 The Law of Cosines
Derivation of the Law of Cosines ▪ Using the Law of Cosines ▪
Heron’s Formula for the Area of a Triangle
Triangle Side Length Restriction
In any triangle, the sum of the lengths of any two sides must be greater
than the length of the remaining side.
Law of Cosines
In any triangle, with sides a, b, and c,
Note - If C = 90°, then cos C = 0, and the formula becomes:
Example 1
USING THE LAW OF COSINES IN AN
APPLICATION (SAS)
A surveyor wishes to find the distance between two
inaccessible points A and B on opposite sides of a lake.
While standing at point C, she finds that AC = 259 m,
BC = 423 m, and angle ACB measures 132°40′. Find
the distance AB.
Example 2
USING THE LAW OF COSINES TO
SOLVE A TRIANGLE (SAS)
Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m.
Caution - If we used the law of sines to find C rather than B, we would not
have known whether C is equal to 81.7° or its supplement, 98.3°.
Example 3
USING THE LAW OF COSINES TO
SOLVE A TRIANGLE (SSS)
Solve triangle ABC if a = 9.47 ft, b = 15.9 ft, and c = 21.1 ft.
Example 4
DESIGNING A ROOF TRUSS (SSS)
Find the measure of angle B in the figure.
Heron’s Area Formula (SSS)
If a triangle has sides of lengths a, b, and c, with semiperimeter
then the area of the triangle is
Heron of Alexandria
Aeolipile
Example 5
USING HERON’S FORMULA TO FIND
AN AREA (SSS)
The distance “as the crow flies” from Los Angeles to New York is 2451 miles,
from New York to Montreal is 331 miles, and from Montreal to Los Angeles is
2427 miles. What is the area of the triangular region having these three cities
as vertices? (Ignore the curvature of Earth.)
8.3 Vectors, Operations, and the
Dot Product
Basic Terminology ▪ Algebraic Interpretation of Vectors ▪
Operations with Vectors ▪ Dot Product and the Angle Between
Vectors
Scalar: The _____________________ of a quantity.
It can be represented by a real number.
A vector in the plane is a directed line segment.
Consider vector OP or __________
O is called the initial point
P is called the terminal point
Magnitude: length of a vector, expressed as __________
Basic Terminology
Two vectors are equal if and only if they have the same __________________
and same _________________.
Vectors OP and PO have the same
magnitude, but opposite directions.
|OP| = |PO|
A=B
C=D
A≠E
A≠F
Sum of Two Vectors


The sum of two vectors is also a vector.
The vector sum A + B is called the resultant.

The sum of a vector v and its opposite –v
has magnitude 0 and is called the zero
vector.

To subtract vector B from vector A, find
the vector sum A + (–B).
Scalar Product of a Vector
The scalar product of a real number k and a vector u is the vector k ∙ u,
with magnitude |k| times the magnitude of u.
Algebraic Interpretation of
Vectors
A vector with its initial point at the origin is called a ______________________
A position vector u with its endpoint at the point (a, b) is written __________
The numbers a and b are the horizontal and
vertical components of vector u.
The positive angle between the x-axis and a
position vector is the direction angle for the
vector.
Magnitude and Direction Angle of a Vector a, b
The magnitude (length) of a vector u = a, b is given by:
The direction angle θ satisfies _______________where a ≠ 0.
Example 1
FINDING MAGNITUDE AND DIRECTION ANGLE
Find the magnitude and direction angle for u = 3, –2.
Horizontal and Vertical Components
The horizontal and vertical components, respectively, of a vector u having
magnitude |u| and direction angle θ are given by
Example 2
FINDING HORIZONTAL AND VERTICAL
COMPONENTS
Vector w has magnitude 25.0 and direction angle 41.7°. Find the horizontal
and vertical components.
Example 3
WRITING VECTORS IN THE FORM a, b
Write each vector in the figure in the form a, b.
Properties of Parallelograms
1. A parallelogram is a quadrilateral whose opposite sides are parallel.
2. The opposite sides and opposite angles of a parallelogram are equal, and
adjacent angles of a parallelogram are supplementary.
3. The diagonals of a parallelogram bisect each other, but do not
necessarily bisect the angles of the parallelogram.
Example 4
FINDING THE MAGNITUDE OF A
RESULTANT
Two forces of 15 and 22 newtons act on a point in the plane. (A newton is a
unit of force that equals .225 lb.) If the angle between the forces is 100°, find
the magnitude of the resultant vector.
Vector Operations
For any real numbers a, b, c, d, and k,
Example 5
PERFORMING VECTOR OPERATIONS
Let u = –2, 1 and v = 4, 3. Find the following.
(a)
u+v
(b)
–2u
(c)
4u – 3v
Unit Vectors
A unit vector is a vector that has magnitude 1.
i = 1, 0
j = 0, 1
Any vector a, b can be expressed in the form
ai + bj using the unit vectors i and j.
Dot Product
The dot product (or inner product) of the two vectors u = a, b and v = c, d is
denoted u ∙ v, read “u dot v,” and is given by
Example 6
Find each dot product.
(a)
2, 3 ∙ 4, –1
(b)
6, 4 ∙ –2, 3
FINDING DOT PRODUCTS
Properties of the Dot Product
For all vectors u, v, and w and real number k,
(a) u ∙ v = v ∙ u
(b) u ∙ (v + w) = u ∙ v + u ∙ w
(c) (u + v) ∙ w = u ∙ w + v ∙ w
(d) (ku) ∙ v = k(u ∙ v) = u ∙ kv
(e) 0 ∙ u = 0
(f) u ∙ u = |u|2
Geometric Interpretation of the Dot Product
If θ is the angle between the two nonzero vectors u and v,
where 0° ≤ θ ≤ 180°, then
Example 7
FINDING THE ANGLE BETWEEN TWO
VECTORS
Find the angle θ between the two vectors u = 3, 4 and v = 2, 1.
Dot Products
For angles θ between 0° and 180°, cos θ is positive, 0, or negative when θ is
less than, equal to, or greater than 90°, respectively.
Note - If a ∙ b = 0 for two nonzero vectors a and b, then cos θ = 0 and θ = 90°.
Thus, a and b are perpendicular or ________________________ vectors.
8.4 Applications of Vectors
The Equilibrant ▪ Incline Applications ▪ Navigation Applications
Sometimes it is necessary to find a vector that will counterbalance a resultant.

This opposite vector is called the ____________________.

The equilibrant of vector u is the vector –u.
Example 1
FINDING THE MAGNITUDE AND DIRECTION OF
AN EQUILIBRANT
Find the magnitude of the equilibrant of forces of 48 newtons and 60 newtons
acting on a point A, if the angle between the forces is 50°. Then find the angle
between the equilibrant and the 48-newton force.
Example 2
FINDING A REQUIRED FORCE
Find the force required to keep a 50-lb wagon from sliding down a ramp inclined
at 20° to the horizontal. (Assume there is no friction.)
Example 3
FINDING AN INCLINE ANGLE
A force of 16.0 lb is required to hold a 40.0 lb lawn
mower on an incline. What angle does the incline
make with the horizontal?
Example 4
APPLYING VECTORS TO A NAVIGATION
PROBLEM
A ship leaves port on a bearing of 28.0° and travels
8.20 mi. The ship then turns due east and travels
4.30 mi. How far is the ship from port? What is its
bearing from port?
Airspeed and Groundspeed
The airspeed of a plane is its speed
relative to the air.
The groundspeed of a plane is its speed
relative to the ground.
The groundspeed of a plane is
represented by the vector sum of the
airspeed and windspeed vectors.
Example 5
APPLYING VECTORS TO A NAVIGATION
PROBLEM
A plane with an airspeed of 192 mph is headed on
a bearing of 121°. A north wind is blowing (from
north to south) at 15.9 mph. Find the groundspeed
and the actual bearing of the plane.
8.5
Trigonometric (Polar) Form of Complex
Numbers; Products and Quotients
The Complex Plane and Vector Representation ▪ Trigonometric
(Polar) Form ▪ Fractals ▪ Products of Complex Numbers in
Trigonometric Form ▪ Quotients of Complex Numbers in
Trigonometric Form
Horizontal axis: real axis
Vertical axis: imaginary axis
Each complex number a + bi determines a unique position
vector with initial point (0, 0) and terminal point (a, b).
The sum of two complex numbers is represented
by the vector that is the resultant of the vectors
corresponding to the two numbers.
(4 + i) + (1 + 3i) = 5 + 4i
Example 1
EXPRESSING THE SUM OF COMPLEX
NUMBERS GRAPHICALLY
Find the sum of 6 – 2i and –4 – 3i. Graph both complex numbers and their
resultant.
Relationships Among x, y, r, and θ.
Trigonometric (Polar) Form of a Complex Number
The expression r(cos θ + i sin θ) is called the trigonometric form
(or polar form) of the complex number x + yi.
The expression cos θ + i sin θ is sometimes abbreviated cis θ.
Using this notation, r(cos θ + i sin θ) is written r cis θ.
The number r is the absolute value (or modulus) of x + yi, and θ is the
argument of x + yi.
Example 2
CONVERTING FROM TRIGONOMETRIC FORM
TO RECTANGULAR FORM
Express 2(cos 300° + i sin 300°) in rectangular form.
Converting From Rectangular Form to Trigonometric Form
Step 1
Step 2
Sketch a graph of the number x + yi in the
complex plane.
Find r by using the equation
Step 3
Find θ by using the equation
choosing the quadrant indicated in Step 1.
Caution
Be sure to choose the correct quadrant for θ by
referring to the graph sketched in Step 1.
Example 3(a) CONVERTING FROM RECTANGULAR
FORM TO TRIGONOMETRIC FORM
Write
in trigonometric form. (Use radian measure.)
Example 3(b) CONVERTING FROM RECTANGULAR
FORM TO TRIGONOMETRIC FORM
Write –3i in trigonometric form. (Use degree measure.)
CONVERTING BETWEEN TRIGONOMETRIC AND
Example 4
RECTANGULAR FORMS USING CALCULATOR
APPROXIMATIONS
Write each complex number in its alternative form, using calculator
approximations as necessary.
(a)
6(cos 115° + i sin 115°)
(b)
5 – 4i
Example 5
DECIDING WHETHER A COMPLEX
NUMBER IS IN THE JULIA SET
The figure shows the fractal called the Julia set.
To determine if a complex number z = a
+ bi belongs to the Julia set, repeatedly
compute the values of:
If the absolute values of any of the resulting complex numbers exceed 2, then
the complex number z is not in the Julia set. Otherwise z is part of this set and
the point (a, b) should be shaded in the graph.
Determine whether each number belongs to the Julia set.
Product Theorem
are any two complex numbers,
then:
In compact form, this is written:
Example 6
USING THE PRODUCT THEOREM
Find the product of 3(cos 45° + i sin 45°) and 2(cos 135° + i sin 135°). Write the
result in rectangular form.
Quotient Theorem
are any two complex numbers, where
In compact form, this is written
Example 7
Find the quotient
USING THE QUOTIENT THEOREM
Write the result in rectangular form.
8.6
De Moivre’s Theorem; Powers
and Roots of Complex Numbers
Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of
Complex Numbers
De Moivre’s Theorem
is a complex number, then
In compact form, this is written
Example 1
Find
FINDING A POWER OF A COMPLEX
NUMBER
and express the result in rectangular form.
nth Root
For a positive integer n, the complex number a + bi is an nth root of
the complex number x + yi if
nth Root Theorem
If n is any positive integer, r is a positive real number, and θ is in
degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly
n distinct nth roots, given by
where
Note - In the statement of the nth root theorem, if θ is in radians, then
Example 2
FINDING COMPLEX ROOTS
Find the two square roots of 4i. Write the roots in rectangular form.
Example 3
Find all fourth roots of
FINDING COMPLEX ROOTS
Write the roots in rectangular form.
Example 3
FINDING COMPLEX ROOTS (continued)
The graphs of the roots lie on a circle with center at the origin and radius 2. The
roots are equally spaced about the circle, 90° apart.
Example 4
SOLVING AN EQUATION BY FINDING
COMPLEX ROOTS
Find all complex number solutions of x5 – i = 0. Graph them as vectors in the
complex plane.
Example 4
SOLVING AN EQUATION BY FINDING
COMPLEX ROOTS (continued)
The graphs of the roots lie on a unit circle. The roots are equally spaced about the
circle, 72° apart.
8.7
Polar Equations and Graphs
Polar Coordinate System ▪ Graphs of Polar Equations ▪
Converting from Polar to Rectangular Equations ▪ Classifying
Polar Equations
The polar coordinate system is based on a point, called the pole, and a
ray, called the polar axis.



Point P has rectangular coordinates (x, y).
Point P can also be located by giving the directed
angle θ from the positive x-axis to ray OP and the
directed distance r from the pole to point P.
The polar coordinates of point P are (r, θ).

If r > 0, then point P lies on the terminal side of θ.

If r < 0, then point P lies on the ray pointing in
the opposite direction of the terminal side of θ, a
distance |r| from the pole.
Rectangular and Polar Coordinates
If a point has rectangular coordinates
(x, y) and polar coordinates (r, ), then these coordinates are
related as follows.
Example 1
PLOTTING POINTS WITH POLAR
COORDINATES
Plot each point by hand in the polar coordinate system. Then, determine
the rectangular coordinates of each point.
(a) P(2, 30°)
Example 1
PLOTTING POINTS WITH POLAR
COORDINATES
Example 1
PLOTTING POINTS WITH POLAR
COORDINATES (continued)
While a given point in the plane can have only one pair of rectangular
coordinates, this same point can have an infinite number of pairs of
polar coordinates.
Example 2
GIVING ALTERNATIVE FORMS FOR
COORDINATES OF A POINT
(a)
Give three other pairs of polar coordinates for the point P(3, 140°).
(b)
Determine two pairs of polar coordinates for the point with the rectangular
coordinates (–1, 1).
Example 3
EXAMINING POLAR AND RECTANGULAR
EQUATION OF LINES AND CIRCLES
For each rectangular equation, give the equivalent polar equation and
sketch its graph.
(a) y = x – 3
Example 3
(b)
EXAMINING POLAR AND RECTANGULAR
EQUATION OF LINES AND CIRCLES
(continued)
Example 4
GRAPHING A POLAR EQUATION
(CARDIOID)
Example 5
GRAPHING A POLAR EQUATION
(ROSE)
Example 6
Graph
.
GRAPHING A POLAR EQUATION
(LEMNISCATE)
Example 7
GRAPHING A POLAR EQUATION
(SPIRAL OF ARCHIMEDES)
Graph r = 2θ, (θ measured in radians).
Example 8
Convert the equation
CONVERTING A POLAR EQUATION TO
A RECTANGULAR EQUATION
to rectangular coordinates and graph.
Classifying Polar Equations
Circles and Lemniscates
Limaçons
Classifying Polar Equations
Rose Curves
2n leaves if n is even,
n≥2
n leaves if n is odd