Applications of Trigonometric Functions

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Applications of Trigonometric Functions

• Chapter 7

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Right Triangle Trigonometry Applications

• Section 7.1

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Trigonometric Functions of Acute Angles

• 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

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Trigonometric Functions of Acute Angles

• Non-right angles in a right triangle must be
  acute (0 lt µ lt 90)
• Pythagorean Theorem a2 b2 c2

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Trigonometric Functions of Acute Angles
These functions will all be positive
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Trigonometric Functions of Acute Angles

• Example.
• Problem Find the exact value of the six
  trigonometric functions of the angle µ
• Answer

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Complementary Angle Theorem

• Complementary angles Two acute angles whose sum
  is a right angle
• In a right triangle, the two acute angles are
  complementary

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Complementary Angle Theorem
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Complementary Angle Theorem

• Cofunctions
• sine and cosine
• tangent and cotangent
• secant and cosecant
• Theorem. Complementary Angle TheoremCofunctions
  of complementary angles are equal

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Complementary Angle Theorem

• Example
• Problem Find the exact value of tan 12 cot
  78 without using a calculator
• Answer

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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

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Solving Right Triangles

• We know
• a2 b2 c2
• 90

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Solving Right Triangles

• Example.
• Problem If b 6 and 65, find a, c and
• Answer

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Solving Right Triangles

• Example.
• Problem If a 8 and b 5, find c, and
• Answer

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Applications of Right Triangles

• Angle of Elevation
• Angle of Depression

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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

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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

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Key Points

• Trigonometric Functions of Acute Angles
• Complementary Angle Theorem
• Solving Right Triangles
• Applications of Right Triangles

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The Law of Sines

• Section 7.2

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Solving Oblique Triangles

• 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

• Convention
• is always the angle opposite side a
• is always the angle opposite side b
• is always the angle opposite side c

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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 with
• Two angles
• One angle and one other side
• The other two sides

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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)

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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

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Solving SAA Triangles

• Example.
• Problem If b 13, 65, and 35, find a,
  c and
• Answer

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Solving ASA Triangles

• Example.
• Problem If c 2, 68, and 40, find a,
  b and
• Answer

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Solving SSA Triangles

• Ambiguous Case
• Information may result in
• One solution
• Two solutions
• No solutions

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Solving SSA Triangles

• Example.
• Problem If a 7, b 9 and 49, find c,
  and
• Answer

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Solving SSA Triangles

• Example.
• Problem If a 5, b 4 and 80, find c,
  and
• Answer

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Solving SSA Triangles

• Example.
• Problem If a 17, b 14 and 25, find c,
  and
• Answer

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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

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Key Points

• Solving Oblique Triangles
• Law of Sines
• Solving SAA Triangles
• Solving ASA Triangles
• Solving SSA Triangles
• Solving Applied Problems

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The Law of Cosines

• Section 7.3

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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

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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
  Theorem
• Take 90

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Solving SAS Triangles

• Example.
• Problem If a 5, c 9, and 25, find b,
  and
• Answer

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Solving SSS Triangles

• Example.
• Problem If a 7, b 4, and c 8, find ,
  and
• Answer

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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

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Key Points

• Law of Cosines
• Solving SAS Triangles
• Solving SSS Triangles
• Solving Applied Problems

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Area of a Triangle

• Section 7.4

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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

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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.

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Area of SAS Triangles

• Example.
• Problem Find the area A of the triangle for
  which a 12, b 15 and 52
• Solution

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Area of SSS Triangles

• Theorem. Herons FormulaThe area A of a
  triangle with sides a, b and c iswhere

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Area of SSS Triangles

• Example.
• Problem Find the area A of the triangle for
  which a 8, b 6 and c 5
• Solution

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Key Points

• Area of a Triangle
• Area of SAS Triangles
• Area of SSS Triangles

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Simple Harmonic Motion Damped Motion Combining
Waves

• Section 7.5

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Simple Harmonic Motion

• Equilibrium (rest) position
• Amplitude Distance from rest position to
  greatest displacement
• Period Length of time to complete one vibration

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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 gt 0
• Assumes no friction or other resistance

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Simple Harmonic Motion

• Simple harmonic motion is related to circular
  motion

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Simple Harmonic Motion

• Theorem. Simple Harmonic MotionAn 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 ! gt 0 are constants, moves with
  simple harmonic motion.The motion has amplitude
  jaj and period

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Simple Harmonic Motion

• Frequency of an object in simple harmonic motion
  Number of oscillations per unit time
• Frequency f is reciprocal of period

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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

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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

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Simple Harmonic Motion

• Example. (cont.)
• (c) Problem What is the time required for one
  oscillation?
• Answer
• (d) Problem What is the frequency?
• Answer

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Damped Motion

• Most physical systems experience friction or
  other resistance

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Damped Motion

• Theorem. Damped MotionThe 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.

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Damped Motion

• Here jaj is the displacement at t 0 and
  is the period under simple harmonic motion (no
  damping).

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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

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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

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Key Points

• Simple Harmonic Motion
• Damped Motion
• Graphing the Sum of Two Functions