Chapter 2 Trigonometric Functions

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Chapter 2Trigonometric Functions

• 2.1 Degrees and Radians
• 2.2 Linear and Angular Velocity
• 2.3 Trigonometric Functions
• Unit Circle Approach
• 2.4 Additional Applications
• 2.5 Exact Values and Properties of
• Trigonometric Functions

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2.1 Degrees and Radians

• Degree and radian measure of angles
• Angles in standard position
• Arc length and area of a sector of a circle

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Radian Measure of Central Angles

• Example
• Find the radian measure of the central angle
  subtended by an arc of 32 cm in a circle of
  radius 8 cm.
• Solution q 32cm/8cm 4 rad

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Radian-Degree Conversion Formulas

• Example
• Find the radian measure of -1.5 rad in terms of p
  and in decimal form to 4 decimal places.
• Solution qd (qr)(180º/p rad)
• (-1.5)(180/p) 270º/p -85.9437º

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Angles in Standard Position
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Sketching angles in Standard Position

• Sketch these angles in standard position
• A. -60º B. 3p/2 rad C. -3p rad D. 405º

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Coterminal Angles

• Angles that differ by an integer multiple of 2 p
  or 360º are coterminal.
• Example
• Are the angles p/3 rad and 2p/3 rad coterminal?
• Solution (-p/3) (2p/3) -3p/3 -p No
• Example
• Are the angles -135º and 225º coterminal?
• Solution -135º - (-225º) -1(360)º Yes

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Area of a Sector of a Circle

• A ½ r2 q, r radius and q central angle
• Example
• In a circle of radius 3 m find the area of the
  sector with central angle 0.4732.
• Solution A ½ 3m2(0.4732) 2.13 m2

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2.2 Linear and Angular Velocity
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Electrical Wind Generator

• This wind generator has propeller blades 5 m
  long. If the blades are rotating at 8 p rad/sec,
  what is the angular velocity of a point on the
  tip of one blade?
• Solution V 5 (8p) 126 m/sec

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2.3 Trigonometric Functions The Unit Circle
Approach

• Definition of Trigonometric Functions
• Calculator Evaluation
• Application
• Summary of Sign Properties

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Trigonometric Functions
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The Unit Circle

• If a point (a,b) lies on the unit circle, then
  the following are true for the angle x associated
  with that point

• sin x b
• cos x a
• tan x b/a (a ? 0)

• csc x 1/b (b ? 0)
• sec x 1/a (a ? 0)
• cot x a/b (b ? 0)

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Evaluating Trigonometric Functions

• Example
• Find the exact values of the 6 trigonometric
  functions for the point (-4, -3)
• The Pythagorean Theorem shows that the distance
  from the point to the origin is 5.

• sin x -3/5
• cos x -4/5
• tan x 3/4

• csc x -5/3
• sec x -5/4
• cot x 4/3

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Using Given Information to Evaluate Trigonometric
Functions

• Example
• Given that the terminal side of an angle is in
  Quadrant IV and cos x 3/5 find the remaining
  trigonometric functions.
• b2 25 9 16, so b 4
• Sin x 4/5, tan x -4/3, csc x -5/4,
• sec x 5/3 and cot x -3/4

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Reciprocal Relationships
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Calculator Evaluation

• Set the calculator in the proper mode for each
  method of evaluating trigonometric functions.
  Use degree mode or radian mode.
• Example
• Find tan 3.472 rad
• Solution tan 3.472 rad .3430
• Example
• Find csc 192º 47 22
• Solution csc 192º 47 22
• 1/ sin 192.7894 -4.517

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Additional Applications

• Modeling light waves and refraction
• Modeling bow waves
• Modeling sonic booms
• High-energy physics Modeling particle energy
• Psychology Modeling perception

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Light Rays
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Reflected Light

• Example
• What is the angle of incidence a that will cause
  a light beam to be totally reflected?
• Solution sin a (sin 90º)1/1.33
• a sin-1 (1/1.33) 48.8º

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2.5 Exact Values and Properties of Trigonometric
Functions

• Exact values of trigonometric functions at
  special angles
• Reference triangles
• Periodic functions
• Fundamental identities

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Special Angles
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Using Special Angles for Points (a,b)

• Example
• Find sec 5p/4
• Solution (a, b) (-1/v2, -1/v2)
• sec 5p/4 1/a -v2
• Example
• Find sin 135º
• Solution (a, b) (-1/v2, 1/v2)
• sin 135º b 1/v2

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Angles on the Unit Circle
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Using Special Angles for Points (a,b)

• Example
• Find sin 7p/6
• Solution (a, b) (-v3/2, -1/2)
• sin 7p/6 b -1/2

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Reference Triangle and Reference Angle
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Reference Triangles and Angles

• Example
• Sketch the reference triangle and find the
  reference angle a for q -315º.
• Solution

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Periodic Functions

• Adding any integer multiple of 2p to x returns
  the same point on the circle.
• sin x sin (x 2p)
• cos x cos (x 2p)
• If sin x 0.7714 then sin(x 2p) 0.7714

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Fundamental Identities

• csc x 1/b 1/sin x
• sec x 1/a 1/cos x
• cot x a/b 1/tan x
• tan x b/a
• sin x / cos x
• cot x a/b
• cos x / sin x
• sin2x cos2x 1

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Use of Identities
Claim
Proof