Trigonometric Functions

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CHAPTER 4

• Trigonometric Functions

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4.1 Angles Radian Measure

• Objectives
• Recognize use the vocabulary of angles
• Use degree measure
• Use radian measure
• Convert between degrees radians
• Draw angles in standard position
• Find coterminal angles
• Find the length of a circular arc
• Use linear angular speed to describe motion on
  a circular path

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Angles

• An angle is formed when two rays have a common
  endpt.
• Standard position one ray lies along the x-axis
  extending toward the right
• Positive angles measure counterclockwise from the
  x-axis
• Negative angles measure clockwise from the x-axis

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Angle Measure

• Degrees full circle 360 degrees
• Half-circle 180 degrees
• Right angle 90 degrees
• Radians one radian is the measure of the central
  angle that intercepts an arc equal in length to
  the length of the radius (we can construct an
  angle of measure 1 radian!)
• Full circle 2 radians
• Half circle radians
• Right angle radians

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Radian Measure

• The measure of the angle in radians is the ratio
  of the arc length to the radius
• Recall half circle 180 degrees radians
• This provides a conversion factor. If they are
  equal, their ratio1, so we can convert from
  radians to degrees (or vice versa) by multiplying
  by this well-chosen one.
• Example convert 270 degrees to radians

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Convert 145 degrees to radians.
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Coterminal angles

• Angles that have rays at the same spot.
• Angle may be positive or negative (move
  counterclockwise or clockwise) (i.e. 70 degree
  angle coterminal to -290 degree angle)
• Angle may go around the circle more than once
  (i.e. 30 degree angle coterminal to 390 degree
  angle)

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Arc length

• Since radians are defined as the central angle
  created when the arc length radius length for
  any given circle, it makes sense to consider arc
  length when angle is measured in radians
• Recall theta (in radians) is the ratio of arc
  length to radius
• Arc length radius x theta (in radians)

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Linear speed Angular speed

• Speed a particle moves along an arc of the circle
  (v) is the linear speed (distance, s, per unit
  time, t)
• Speed which the angle is changing as a particle
  moves along an arc of the circle is the angular
  speed.(angle measure in radians, per unit time, t)

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Relationship between linear speed angular speed

• Linear speed is the product of radius and angular
  speed.
• Example The minute hand of a clock is 6 inches
  long. How fast is the tip of the hand moving?
• We know angular speed 2 pi per 60 minutes

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4.2 Trigonometric Functions The Unit Circle

• Objectives
• Use a unit circle to define trigonometric
  functions of real numbers
• Recognize the domain range of sine cosine
• Find exact values of the trig. functions at pi/4
• Use even odd trigonometric functions
• Recognize use fundamental identities
• Use periodic properties
• Evaluate trig. functions with a calculator

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What is the unit circle?

• A circle with radius 1 unit
• Why are we interested in this circle? It
  provides convenient (x,y) values as we work our
  way around the circle.
• (1,0), theta 0
• (0,1), theta pi/2
• (-1,0), theta pi
• (0,-1), theta 3 pi/2
• ALSO, any (x,y) point on the circle would be at
  the end of the hypotenuse of a right triangle
  that extends from the origin, such that

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sin t and cos t

• For any point (x,y) found on the unit circle,
  xcos t and ysin t
• t any real number, corresponding to the arc
  length of the unit circle
• Example at the point (1,0), the cos t 1 and
  sin t 0. What is t? t is the arc length at
  that point AND since its a unit circle, we know
  the arc length central angle, in radians.
  THUS, cos (0) 1 and sin (0)0

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Relating all trigonometric functions to sin t and
cos t
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Pythagorean Identities

• Every point (x,y) on the unit circle corresponds
  to a real number, t, that represents the arc
  length at that point
• Since and x cos(t)
  and ysin(t), then
• If each term is divided by , the
  result is
• If each term is divided by , the
  result is

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Given csc t 13/12, find the values of the other
6 trig. functions of t

• sin t 12/13 (reciprocal)
• cos t 5/13 (Pythagorean)
• sec t 13/5 (reciprocal)
• tan t 12/5 (sin(t)/cos(t))
• cot t 5/12 (reciprocal)

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Trig. functions are periodic

• sin(t) and cos(t) are the (x,y) coordinates
  around the unit circle and the values repeat
  every time a full circle is completed
• Thus the period of both sin(t) and cos(t) 2 pi
• sin(t)sin(2pi t) cos(t)cos(2pi t)
• Since tan(t) sin(t)/cos(t), we find the values
  repeat (become periodic) after pi, thus
  tan(t)tan(pi t)

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4.3 Right Triangle Trigonometry

• Objectives
• Use right triangles to evaluate trig. Functions
• Find function values for 30 degrees, 45 degrees
  60 degrees
• Use equal cofunctions of complements
• Use right triangle trig. to solve applied problems

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Within a unit circle, and right triangle can be
sketched

• The point on the circle is (x,y) and the
  hypotenuse 1. Therefore, the x-value is the
  horizontal leg and the y-value is the vertical
  leg of the right triangle formed.
• cos(t)x which equals x/1, therefore the cos
  (t)horizontal leg/hypotenuse adjacent
  leg/hypotense
• sin(t)y which equals y/1, therefore the sin(t)
  vertical leg/hypotenuse opposite leg/hypotenuse

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The relationships holds true for ALL right
triangles (other 3 trig. functions are found as
reciprocals)
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Find the value of 6 trig. functions of the angles
in a right triangle.

• Given 2 sides, the value of the 3rd side can be
  found, using Pythagorean theorem
• After side lengths of all 3 sides is known, find
  sin as opposite/hypotenuse
• cos adjacent/hypotenuse
• tan opposite/adjacent
• csc 1/sin
• sec 1/cos
• cot 1/tan

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Given a right triangle with hypotenuse 5 and
side adjacent angle B of length2, find tan B
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Special Triangles

• 30-60 right triangle, ratio of sides of the
  triangle is 12 , 2 (longest) is the length
  of the hypotenuse, the shortest side (opposite
  the 30 degree angle) is 1 and the remaining side
  (opposite the 60 degree angle) is
• 45-45 right triangle The 2 legs are the same
  length since the angles opposite them are equal,
  thus 11. Using pythagorean theorem, the
  remaining side, the hypotenuse, is

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Cofunction Identities

• Cofunctions are those that are the reciprocal
  functions (cofunction of tan is cot, cofunction
  of sin is cos, cofunction of sec is csc)
• For an acute angle, A, of a right triangle, the
  side opposite A would be the side adjacent to the
  other acute angle, B
• Therefore sin A cos B
• Since A B are the acute angles of a right
  triangle, their sum 90 degrees, thus B
• function(A)cofunction

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4.4 Trigonometric Functions of Any Angle

• Objectives
• Use the definitions of trigonometric functions of
  any angle
• Use the signs of the trigonometric functions
• Find reference angles
• Use reference angles to evaluate trigonometric
  functions

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Trigonometric functions of Any Angle

• Previously, we looked at the 6 trig. functions of
  angles in a right triangle. These angles are all
  acute. What about negative angles? What about
  obtuse angles?
• These angles exist, particularly as we consider
  moving around a circle
• At any point on the circle, we can drop a
  vertical line to the x-axis and create a
  triangle. Horizontal side x, vertical sidey,
  hypotenuser.

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Trigonometric Functions of Any Angle (continued)

• If, for example, you have an angle whose terminal
  side is in the 3rd quadrant, then the x y
  values are both negative. The radius, r, is
  always a positive value.
• Given a point (-3,-4), find the 6 trig. functions
  associated with the angle formed by the ray
  containing this point.
• x-3, y-4, r
• (continued next slide)

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Example continued

• sin A -4/5, cos A -3/5, tan A 4/3
• csc A -5/4, sec A -5/3, cot A ¾
• Notice that the same values of the trig.
  functions for angle A would be true for the
  angles 360A, A-360 (negative values)

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Examining the 4 quadrants

• Quadrant I x y are positive
• all 6 trig. functions are positive
• Quadrant II x negative, y positive
• positive sin, csc negative cos, sec, tan,
  cot
• Quadrant III x negative, y negative
• positive tan, cot negative sin, csc, cos,
  sec
• Quadrant IV x positive, y negative
• positive cos, sec negative sin, csc, cot,
  tan

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Reference angles

• Angles in all quadrants can be related to a
  reference angle in the 1st quadrant
• If angle A is in quadrant II, its related angle
  in quad I is 180-A. The numerical values of the
  6 trig. functions will be the same, except the x
  (cos, sec, tan, cot) will all be negative
• If angle A is in quad III, its related angle in
  quad I is 180A. Now x y are both neg, so sin,
  csc, cos, sec are all negative.

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Reference angles cont.

• If angle A is in quad IV, the reference angle is
  360-A. The y value is negative, so the sin, csc,
  tan cot are all negative.

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Special angles

• We often work with the special angles of the
  special triangles. Its good to remember them
  both in radians degrees
• If you know the trig. functions of the special
  angles in quad I, you know them in every
  quadrant, by determining whether the x or y is
  positive or negative

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4.5 Graphs of Sine Cosine

• Objectives
• Understand the graph of y sin x
• Graph variations of y sin x
• Understand the graph of y cos x
• Graph variations of y cos x
• Use vertical shifts of sin cosine curves
• Model periodic behavior

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Graphing y sin x

• If we take all the values of sin x from the unit
  circle and plot them on a coordinate axis with x
  angles and y sin x, the graph is a curve
• Range -1,1
• Domain (all reals)

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Graphing y cos x

• Unwrap the unit circle, and plot all x values
  from the circle (the cos values) and plot on the
  coordinate axes, x angle measures (in radians)
  and y cos x
• Range -1,1
• Domain (all reals)

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Comparisons between ycos x and ysin x

• Range Domain SAME
• range -1,1, domain (all reals)
• Period SAME (2 pi)
• Intercepts Different
• sin x crosses through origin and intercepts the
  x-axis at all multiples of
• cos x intercepts y-axis at (0,1) and intercepts
  x-axis at all odd multiples of

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Amplitude Period

• The amplitude of sin x cos x is 1. The greatest
  distance the curves rise fall from the axis is
  1.
• The period of both functions is 2 pi. This is
  the distance around the unit circle.
• Can we change amplitude? Yes, if the function
  value (y) is multiplied by a constant, that is
  the NEW amplitude, example y 3 sin x

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Amplitude Period (cont)

• Can we change the period? Yes, the length of the
  period is a function of the x-value.
• Example y sin(3x)
• The amplitude is still 1. (Range -1,1)
• Period is

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Phase Shift

• The graph of ysin x is shifted left or right
  of the original graph
• Change is made to the x-values, so its
  addition/subtraction to x-values.
• Example y sin(x- ), the graph of ysin x
  is shifted right

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Vertical Shift

• The graph ysin x can be shifted up or down on
  the coordinate axis by adding to the y-value.
• Example
• y sin x 3 moves the graph of sin x up 3
  units.

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Graph y 2cos(x- ) - 2

• Amplitude 2
• Phase shift right
• Vertical shift down 2

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4.6 Graphs of Other Trigonometric Functions

• Objectives
• Understand the graph of y tan x
• Graph variations of y tan x
• Understand the graph of y cot x
• Graph variations of y cot x
• Understand the graphs of y csc x and y sec x

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y tan x

• Going around the unit circle, where the y value
  is 0, (sin x 0), the tangent is undefined.
• At x the graph of y tan
  x has vertical asymptotes
• x-intercepts where cos x 0, x
  

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Characteristics of y tan x

• Period
• Domain (all reals except odd multiples of
• Range (all reals)
• Vertical asymptotes odd multiples of
• x intercepts all multiples of
• Odd function (symmetric through the origin, quad
  I mirrors to quad III)

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Transformations of y tan x

• Shifts (vertical phase) are done as the shifts
  to y sin x
• Period change (same as to ysin x, except the
  original period of tan x is pi, not 2 pi)

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Graph y -3 tan (2x) 1

• Period is now pi/2
• Vertical shift is up 1
• -3 impacts the amplitude
• Since tan x has no amplitude, we consider the
  point ½ way between intercept asymptote, where
  the y-value1. Now the y-value at that point is
  -3.
• See graph next slide.

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Graph y -3 tan (2x) 1
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Graphing y cot x

• Vertical asymptotes are where sin x 0,
  (multiples of pi)
• x-intercepts are where cos x 0 (odd multiples
  of pi/2)

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y csc x

• Reciprocal of y sin x
• Vertical tangents where sin x 0 (x integer
  multiples of pi)
• Range
• Domain all reals except integer multiples of pi
• Graph on next slide

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Graph of y csc x
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y sec x

• Reciprocal of y cos x
• Vertical tangents where cos x 0 (odd multiples
  of pi/2)
• Range
• Domain all reals except odd multiples of pi/2
• Graph next page

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Graph of y sec x
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4.7 Inverse Trigonometric Functions

• Objectives
• Understand the use the inverse sine function
• Understand and use the inverse cosine function
• Understand and use the inverse tangent function
• Use a calculator to evaluate inverse trig.
  functions
• Find exact values of composite functions with
  inverse trigonometric functions

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What is the inverse sin of x?

• It is the ANGLE (or real ) that has a sin value
  of x.
• Example the inverse sin of ½ is pi/6 (arcsin ½
  pi/6)
• Why? Because the sin(pi/6) ½
• Shorthand notation for inverse sin of x is arcsin
  x or
• Recall that there are MANY angles that would have
  a sin value of ½. We want to be consistent and
  specific about WHICH angle were referring to, so
  we limit the range to (quad I
  IV)

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Find the domain of y

• The domain of any function becomes the range of
  its inverse, and the range of a function becomes
  the domain of its inverse.
• Range of y sin x is -1,1, therefore the
  domain of the inverse sin (arcsin x) function is
  -1,1

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Trigonometric values for special angles

• If you know sin(pi/2) 1, you know the inverse
  sin(1) pi/2
• KNOW TRIG VALUES FOR ALL SPECIAL ANGLES (once you
  do, you know the inverse trigs as well!)

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Find
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Graph y arcsin (x)
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The inverse cosine function

• The inverse cosine of x refers to the angle (or
  number) that has a cosine of x
• Inverse cosine of x is represented as arccos(x)
  or
• Example arccos(1/2) pi/3 because the cos(pi/3)
  ½
• Domain -1,1
• Range 0,pi (quadrants I II)

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Graph y arccos (x)
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The inverse tangent function

• The inverse tangent of x refers to the angle (or
  number) that has a tangent of x
• Inverse tangent of x is represented as arctan(x)
  or
• Example arctan(1) pi/4 because the tan(pi/4)1
  
• Domain (all reals)
• Range -pi/2,pi/2 (quadrants I IV)

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Graph y arctan(x)
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Evaluating compositions of functions their
inverses

• Recall The composition of a function and its
  inverse x. (what the function does, its
  inverse undoes)
• This is true for trig. functions their
  inverses, as well ( PROVIDED x is in the range of
  the inverse trig. function)
• Example arcsin(sin pi/6) pi/6, BUT arcsin(sin
  5pi/6) pi/6
• WHY? 5pi/6 is NOT in the range of arcsin x, but
  the angle that has the same sin in the
  appropriate range is pi/6

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4.8 Applications of Trigonometric Functions

• Objectives
• Solve a right triangle.
• Solve problems involving bearings.
• Model simple harmonic motion.

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Solving a Right Triangle

• This means find the values of all angles and all
  side lengths.
• Sum of angles 180 degrees, and if one is a
  right angle, the sum of the remaining angles is
  90 degrees.
• All sides are related by the Pythagorean Theorem
• Using ratio definition of trig functions (sin x
  opposite/hypotenuse, tan x opposite/adjacent,
  cos x adjacent/hypotenuse), one can find
  remaining sides if only one side is given

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Example A right triangle has an hypotenuse 6
cm with an angle 35 degrees. Solve the
triangle.

• cos(35 degrees) .819 (using calculator)
• cos(35 degrees) adjacent/6 cm
• Thus, .819 adjacent/6 cm, adjacent 4.9 cm
• Remaining angle 55 degrees
• Remaining side

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Trigonometry Bearings

• Bearings are used to describe position in
  navigation and surveying. Positions are
  described relative to a NORTH or SOUTH axis
  (y-axis). (Different than measuring from the
  standard position, the positive x-axis.)
• means the direction is 55 degrees
  from the north toward the east (in quadrant I)
• means the direction is 35 degrees
  from the south toward the west (in quadrant III)

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