applications of trigonometric functions chapter 7 texpoint fonts used in emf. read the texpoint...
Post on 18-Dec-2015
222 Views
Preview:
TRANSCRIPT
Applications of Trigonometric FunctionsChapter 7
Right Triangle Trigonometry; ApplicationsSection 7.1
Trigonometric Functions of Acute
AnglesRight 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
AnglesNon-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
AnglesExample.
Problem: Find the exact value of the six trigonometric functions of the angle µ
Answer:
Complementary Angle Theorem
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
Cofunctions: sine and cosinetangent and cotangentsecant and cosecant
Theorem. [Complementary Angle Theorem]Cofunctions of complementary angles are equal
Complementary Angle Theorem
ExampleProblem: Find the exact value of
tan 12± { cot 78± without using a calculator
Answer:
Solving Right Triangles
Convention: ® 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: Express lengths rounded to two
decimal places Express angles in degrees rounded to
one decimal place
Solving Right Triangles
We know:a2 + b2 = c2
® + ¯ = 90±
Solving Right Triangles
Example. Problem: If b = 6 and ¯ = 65±,
find a, c and ®Answer:
Solving Right Triangles
Example. Problem: If a = 8 and b = 5, find
c, ® and ¯Answer:
Applications of Right Triangles
Angle of ElevationAngle of Depression
Applications of Right Triangles
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
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
Trigonometric Functions of Acute Angles
Complementary Angle Theorem
Solving Right TrianglesApplications of Right
Triangles
The Law of Sines
Section 7.2
Solving Oblique Triangles
Oblique Triangle: A triangle which is not a right triangleCan have three acute angles,
orTwo acute angles and one
obtuse angle (an angle between 90± and 180±)
Solving Oblique Triangles
Convention: ® is always the angle opposite
side a ¯ is always the angle opposite
side b ° is always the angle opposite
side c
Solving Oblique Triangles
Solving an oblique triangle: Finding the missing lengths of the sides and missing measures of the angles
Must know one side, together withTwo anglesOne angle and one other sideThe other two sides
Solving Oblique Triangles
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)
Law of Sines Theorem. [Law of Sines]
For a triangle with sides a, b, c and opposite angles ®, ¯, °, respectively
Law of Sines can be used to solve ASA, SAA and SSA triangles
Use the fact that ® + ¯ + ° = 180±
Solving SAA Triangles
Example. Problem: If b = 13, ® = 65±, and
¯ = 35±, find a, c and °Answer:
Solving ASA Triangles
Example. Problem: If c = 2, ® = 68±, and ¯
= 40±, find a, b and °Answer:
Solving SSA Triangles
Ambiguous CaseInformation may result in
One solutionTwo solutionsNo solutions
Solving SSA Triangles
Example. Problem: If a = 7, b = 9 and ¯ =
49±, find c, ® and °Answer:
Solving SSA Triangles
Example. Problem: If a = 5, b = 4 and ¯ =
80±, find c, ® and °Answer:
Solving SSA Triangles
Example. Problem: If a = 17, b = 14 and ¯
= 25±, find c, ® and °Answer:
Solving Applied Problems
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
Solving Oblique TrianglesLaw of SinesSolving SAA TrianglesSolving ASA TrianglesSolving SSA TrianglesSolving Applied Problems
The Law of Cosines
Section 7.3
Law of Cosines
Theorem. [Law of Cosines]For a triangle with sides a, b, c and opposite angles ®, ¯, °, respectively
Law of Cosines can be used to solve SAS and SSS triangles
Law of Cosines
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 TheoremTake ° = 90±
Solving SAS Triangles
Example. Problem: If a = 5, c = 9, and ¯ =
25±, find b, ® and °Answer:
Solving SSS Triangles
Example. Problem: If a = 7, b = 4, and c =
8, find ®, ¯ and °Answer:
Solving Applied Problems
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
Law of CosinesSolving SAS TrianglesSolving SSS TrianglesSolving Applied Problems
Area of a Triangle
Section 7.4
Area of a Triangle
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
If we know two sides a and b and the included angle °, then
Also,
Theorem.The area A of a triangle equals one-half the product of two of its sides times the sine of their included angle.
Area of SAS Triangles
Example.Problem: Find the area A of the
triangle for which a = 12, b = 15 and ° = 52±
Solution:
Area of SSS Triangles
Theorem. [Heron’s Formula]The area A of a triangle with sides a, b and c is
where
Area of SSS Triangles
Example.Problem: Find the area A of the
triangle for which a = 8, b = 6 and c = 5
Solution:
Key Points
Area of a TriangleArea of SAS TrianglesArea of SSS Triangles
Simple Harmonic Motion; Damped Motion; Combining Waves Section 7.5
Simple Harmonic Motion
Equilibrium (rest) position
Amplitude: Distance from rest position to greatest displacement
Period: Length of time to complete one vibration
Simple Harmonic Motion
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
Simple harmonic motion is related to circular motion
Simple Harmonic Motion
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
Frequency of an object in simple harmonic motion: Number of oscillations per unit time
Frequency f is reciprocal of period
Simple Harmonic Motion
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
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
Example. (cont.)
(c) Problem: What is the time
required for one oscillation?
Answer:
(d) Problem: What is the
frequency?
Answer:
Damped Motion
Most physical systems experience friction or other resistance
Damped Motion
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 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:
2
32
22
-6
-4
-2
2
4
6
Graphing the Sum of Two Functions
Example. f(x) = x + cos(2x)Problem: Use the method of
adding y-coordinates to graph y = f(x)
Answer:
Key Points
Simple Harmonic MotionDamped MotionGraphing the Sum of Two
Functions
top related