Transcript Trigonometric Functions - University of Wisconsin

Applications of
Trigonometric
Functions
Chapter 7
Right Triangle
Trigonometry;
Applications
Section 7.1
Trigonometric Functions of
Acute Angles
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Right triangle: Triangle in which one
angle is a right angle
Hypotenuse: Side opposite the right
angle in a right triangle
Legs: Remaining two sides in a right
triangle
Trigonometric Functions of
Acute Angles
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Non-right angles in a right triangle
must be acute (0± < µ < 90±)
Pythagorean Theorem: a2 + b2 = c2
Trigonometric Functions of
Acute Angles
These functions will
all be positive
Trigonometric Functions of
Acute Angles
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Example.
Problem: Find the exact value of the six
trigonometric functions of the angle µ
Answer:
Complementary Angle
Theorem
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Complementary angles: Two acute
angles whose sum is a right angle
In a right triangle, the two acute
angles are complementary
Complementary Angle
Theorem
Complementary Angle
Theorem
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Cofunctions:
sine and cosine
 tangent and cotangent
 secant and cosecant
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Theorem. [Complementary Angle
Theorem]
Cofunctions of complementary angles
are equal
Complementary Angle
Theorem
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Example
Problem: Find the exact value of
tan 12± { cot 78± without using a
calculator
Answer:
Solving Right Triangles
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Convention:
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® is always the angle opposite side a
¯ is always the angle opposite side b
Side c is the hypotenuse
Solving a right triangle: Finding the
missing lengths of the sides and missing
measures of the angles
Convention:
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Express lengths rounded to two decimal places
Express angles in degrees rounded to one
decimal place
Solving Right Triangles
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We know:
a2 + b2 = c 2
 ® + ¯ = 90±
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Solving Right Triangles
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Example.
Problem: If b = 6 and ¯ = 65±, find a, c
and ®
Answer:
Solving Right Triangles
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Example.
Problem: If a = 8 and b = 5, find c, ® and
¯
Answer:
Applications of Right Triangles
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Angle of Elevation
Angle of Depression
Applications of Right Triangles
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Example.
Problem: The angle of elevation of the Sun
is 35.1± at the instant it casts a shadow
789 feet long of the Washington
Monument. Use this information to
calculate the height of the monument.
Answer:
Applications of Right Triangles
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Direction or Bearing from a point O
to a point P : Acute angle µ between
the ray OP and the vertical line
through O
Key Points
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Trigonometric Functions of Acute
Angles
Complementary Angle Theorem
Solving Right Triangles
Applications of Right Triangles
The Law of Sines
Section 7.2
Solving Oblique Triangles
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Oblique Triangle: A triangle which is
not a right triangle
Can have three acute angles, or
 Two acute angles and one obtuse angle
(an angle between 90± and 180±)
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Solving Oblique Triangles
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Convention:
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® is always the angle opposite side a
¯ is always the angle opposite side b
° is always the angle opposite side c
Solving Oblique Triangles
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Solving an oblique triangle: Finding
the missing lengths of the sides and
missing measures of the angles
Must know one side, together with
Two angles
 One angle and one other side
 The other two sides
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Solving Oblique Triangles
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Known information:
One side and two angles: (ASA, SAA)
 Two sides and angle opposite one of
them: (SSA)
 Two sides and the included angle (SAS)
 All three sides (SSS)
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Law of Sines
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Theorem. [Law of Sines]
For a triangle with sides a, b, c and
opposite angles ®, ¯, °, respectively
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Law of Sines can be used to solve
ASA, SAA and SSA triangles
Use the fact that ® + ¯ + ° = 180±
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Solving SAA Triangles
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Example.
Problem: If b = 13, ® = 65±, and ¯ = 35±,
find a, c and °
Answer:
Solving ASA Triangles
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Example.
Problem: If c = 2, ® = 68±, and ¯ = 40±,
find a, b and °
Answer:
Solving SSA Triangles
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Ambiguous Case
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Information may result in
One solution
 Two solutions
 No solutions
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Solving SSA Triangles
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Example.
Problem: If a = 7, b = 9 and ¯ = 49±, find
c, ® and °
Answer:
Solving SSA Triangles
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Example.
Problem: If a = 5, b = 4 and ¯ = 80±, find
c, ® and °
Answer:
Solving SSA Triangles
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Example.
Problem: If a = 17, b = 14 and ¯ = 25±,
find c, ® and °
Answer:
Solving Applied Problems
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Example.
Problem: An airplane is sighted at the
same time by two ground observers who
are 5 miles apart and both directly west
of the airplane. They report the angles of
elevation as 12± and 22±. How high is the
airplane?
Solution:
Key Points
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Solving Oblique Triangles
Law of Sines
Solving SAA Triangles
Solving ASA Triangles
Solving SSA Triangles
Solving Applied Problems
The Law of
Cosines
Section 7.3
Law of Cosines
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Theorem. [Law of Cosines]
For a triangle with sides a, b, c and
opposite angles ®, ¯, °, respectively
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Law of Cosines can be used to solve
SAS and SSS triangles
Law of Cosines
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Theorem. [Law of Cosines - Restated]
The square of one side of a triangle
equals the sum of the squares of the
two other sides minus twice their
product times the cosine of the
included angle.
The Law of Cosines generalizes the
Pythagorean Theorem
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Take ° = 90±
Solving SAS Triangles
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Example.
Problem: If a = 5, c = 9, and ¯ = 25±,
find b, ® and °
Answer:
Solving SSS Triangles
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Example.
Problem: If a = 7, b = 4, and c = 8, find
®, ¯ and °
Answer:
Solving Applied Problems
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Example. In flying the 98 miles from
Stevens Point to Madison, a student pilot
sets a heading that is 11± off course and
maintains an average speed of 116 miles
per hour. After 15 minutes, the instructor
notices the course error and tells the
student to correct the heading.
(a) Problem: Through what angle will the plane
move to correct the heading?
Answer:
(b) Problem: How many miles away is Madison
when the plane turns?
Answer:
Key Points
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Law of Cosines
Solving SAS Triangles
Solving SSS Triangles
Solving Applied Problems
Area of a
Triangle
Section 7.4
Area of a Triangle
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Theorem.
The area A of a triangle is
where b is the base and h is an
altitude drawn to that base
Area of SAS Triangles
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If we know two sides a and b and the
included angle °, then
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Also,
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Theorem.
The area A of a triangle equals onehalf the product of two of its sides
times the sine of their included angle.
Area of SAS Triangles
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Example.
Problem: Find the area A of the triangle
for which a = 12, b = 15 and ° = 52±
Solution:
Area of SSS Triangles
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Theorem. [Heron’s Formula]
The area A of a triangle with sides a,
b and c is
where
Area of SSS Triangles
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Example.
Problem: Find the area A of the triangle
for which a = 8, b = 6 and c = 5
Solution:
Key Points
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Area of a Triangle
Area of SAS Triangles
Area of SSS Triangles
Simple Harmonic
Motion; Damped
Motion; Combining
Waves
Section 7.5
Simple Harmonic Motion
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Equilibrium (rest)
position
Amplitude:
Distance from rest
position to greatest
displacement
Period: Length of
time to complete
one vibration
Simple Harmonic Motion
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Simple harmonic motion:
Vibrational motion in which
acceleration a of the object is directly
proportional to the negative of its
displacement d from its rest position
a = {kd, k > 0
Assumes no friction or other
resistance
Simple Harmonic Motion
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Simple harmonic motion is related to
circular motion
Simple Harmonic Motion
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Theorem. [Simple Harmonic Motion]
An object that moves on a coordinate
axis so that the distance d from its
rest position at time t is given by
either
d = a cos(!t) or d = a sin(!t)
where a and ! > 0 are constants,
moves with simple harmonic motion.
The motion has amplitude jaj and
period
Simple Harmonic Motion
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Frequency of an object in simple
harmonic motion: Number of
oscillations per unit time
Frequency f is reciprocal of period
Simple Harmonic Motion
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Example. Suppose that an object attached
to a coiled spring is pulled down a distance
of 6 inches from its rest position and then
released.
Problem: If the time for one oscillation is 4
seconds, write an equation that relates the
displacement d of the object from its rest
position after time t (in seconds). Assume no
friction.
Answer:
Simple Harmonic Motion
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Example. Suppose that the displacement d
(in feet) of an object at time t (in seconds)
satisfies the equation
d = 6 sin(3t)
(a) Problem: Describe the motion of the object.
Answer:
(b) Problem: What is the maximum displacement
from its resting position?
Answer:
Simple Harmonic Motion
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Example. (cont.)
(c) Problem: What is the time required for
one oscillation?
Answer:
(d) Problem: What is the frequency?
Answer:
Damped Motion
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Most physical systems experience
friction or other resistance
Damped Motion
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Theorem. [Damped Motion]
The displacement d of an oscillating object
from its at-rest position at time t is given
by
where b is a damping factor (damping
coefficient) and m is the mass of the
oscillating object.
Damped Motion
Here jaj is the displacement at t = 0
and
is the period under simple
harmonic motion (no damping).
Damped Motion
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Example. A simple pendulum with a
bob of mass 15 grams and a damping
factor of 0.7 grams per second is pulled
11 centimeters from its at-rest position
and then released. The period of the
pendulum without the damping effect is
3 seconds.
Problem: Find an equation that describes the
position of the pendulum bob.
Answer:
Graphing the Sum of Two
Functions
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Example. f(x) = x + cos(2x)
Problem: Use the method of adding ycoordinates to graph y = f(x)
Answer:
6
4
2
3
2
2
-2
-4
-6
2
2
Key Points
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Simple Harmonic Motion
Damped Motion
Graphing the Sum of Two Functions