Chapter 4 – Trigonometric Functions. 4.1 – Angles and Their Measures Degrees and Radians Degree...

Chapter 4 – Trigonometric Chapter 4 – Trigonometric FunctionsFunctions

4.1 – Angles and Their Measures4.1 – Angles and Their Measures

Degrees and RadiansDegrees and RadiansDegree – denoted � , is a unit of

angular measure equal to 1/180th of a straight angle.

In the DMS (____________________) system of angular measure, each degree is subdivided into 60 minutes (denoted by ’) and each minute is subdivided into 60 seconds (denoted by ’’ ).

Working with DMS Working with DMS measuremeasure(a) Convert 37.425 � to DMS

(b) Convert 42 �24’36’’ to degrees.

RadianRadianA central angle of a circle has

measure 1 radian if it intercepts an arc with the same length as the radius.

Working with Radian Working with Radian MeasureMeasure(a) How many radians are in 90

degrees?

(b)How many degrees are in /3 radians?

Cont…Cont…(a) Find the length of an arc

intercepted by a central angle of ½ radian in a circle of radius 5 inches.

Degree-Radian ConversionDegree-Radian ConversionTo convert radians to degrees,

multiply by

To convert degrees to radians, multiply by

Practice:Practice:Work with partners

◦Pg 356 # 1-24e

Do Now:Convert from radians to degrees:

◦π/6

◦π/10

◦5π/9

Circular Arc LengthArc Length Formula (Radian Measure)

◦ If θ is a central angle in a circle of radius r, and if θ is measure in radians, then the length s of the intercepted arc is given by S=r θ

Arc Length Formula (Degree Measure)◦ If θ is a central angle in a circle of radius r,

and if θ is measured in degrees, then the length s of the intercepted arc is given by S=πr θ /180

Find the perimeter of a 60° slice of a large pizza (with 8 in radius)

4.2 - Trigonometric Functions of Acute Angles

HW: Pg 368 #2-32e

Right Triangle Trigonometry - DEF: Trigonometric FunctionsLet be an acute angle in the

right ABC. Then◦Sine θ = sin θ = ◦Cosine θ = cos θ =◦Tangent θ = tan θ =◦Cosecant θ = csc θ = hyp/opp◦Secant θ = sec θ = hyp/adj◦Cotangent θ = cot θ = adj/opp

Find the values of all six trigonometric functions for an angle of 45°

Find the values of all six trigonometric functions for an angle of 30º

Let θ be an acute angle such that sin θ θ = 5/6. Evaluate the other five trigonometric functions of θθ.

Solving a Right TriangleSolving a Right TriangleA right triangle with a hypotenuse of

8 includes a 37 � angle. Find the measures of the other two angles and the lengths of the other two sides.

From a point 340 feet away from From a point 340 feet away from the base of the Tower of the base of the Tower of Weehawken, the angle of elevation Weehawken, the angle of elevation

to the top of the building is 65 to the top of the building is 65 ��..Find the height h of the building.

4.3 - Trigonometry Extended - The Circular Functions

HW: Pg. 381 #2-6e, 26-36e

Trigonometric Functions of Any Angle

Positive angles are generated by counterclockwise rotations

Negative angles are generated by clockwise rotations.

Standard Position

Vertex -

Initial side -

Coterminal AnglesTwo angles can have the same

initial side and the same terminal side, yet have different measures - called coterminal angles.

Find Coterminal Angles

(a) 40 �

(b) -160 �

(c) 2/3 radians

Trigonometry in Quadrant I:

Let P(x,y) be any point in the first quadrant, let r be the distance from P to the origin.

1. Use the acute angle definition to show the sinθ = y/r

2. Express cos θ= in terms of x and r.

3. Express tan θ= in terms of x and y.

4. Express the remaining three basic trigonometric functions in terms of x, y, and r.

P(x,y)

x

y

θ

r

Evaluating Trig Functions Determined by a Point in QILet θ be the acute angle in

standard position whose terminal side contains the point (4,3). Find the six trigonometric functions of θ.

Let θ be an angle in standard position whose terminal side contains the point (-5,2). Find the six trigonometric functions of θ.

Evaluating Trig Functions Determined by a Point NOT in QI

Trigonometric Functions of Trigonometric Functions of any Angleany AngleLet θ be any angle in standard position and

let P(x,y) be any point on the terminal side of the angle (except the origin). Let r denote the distance from P(x,y) to the origin, i.e., Let r = √(x2+y2). Then◦ Sin θ= ◦ cos θ=◦ tan θ=◦ csc θ=◦ sec θ=◦ cot θ=

P(x,y) y

x

r

θ

Evaluating the Trig Functions Evaluating the Trig Functions of 315 of 315 ��Find the six trigonometric

functions of 315 �

Evaluating More Trig Evaluating More Trig FunctionsFunctionsSin(-210 �)

Tan(5/3)

Sec(-3/4)

……Sin(-270 �)

Tan(3)

Sec(11/2)

When are sin, cos, and tan When are sin, cos, and tan positive?positive?Quadrant 1-

Quadrant II –

Quadrant III –

Quadrant IV –

Using One Ratio to Find the Using One Ratio to Find the OthersOthers

Find cosFind cos θ θ and tan and tan θ θ if: if:Sinθ = 3/7 and tanθ <0

sec θ=3 and sin θ>0

cot θ is undefined and sec θ is negative

……Try on your ownTry on your ownFind sin θ and tan θ if cos θ=2/3

and cot θ>0

Find tan θ and sec θ if sin θ=-2/5 and cos θ>0

Find sec θ and csc θ if tan θ=-4/3 and sin θ>0

Trigonometric Functions of Trigonometric Functions of Real NumbersReal NumbersDEF: Unit Circle

◦The unit circle is a circle of radius 1 centered at the origin.

Wrapping FunctionWrapping FunctionConnects points on a number line

with points on the circle

Trigonometric Functions of Trigonometric Functions of Real NumbersReal NumbersLet t be any real number, and let P(x,y) be the

point corresponding to t when the number line is wrapped onto the unit circle as described above. Then ◦ Sin t = ◦ Cos t =◦ Tan t =◦ Csc t =◦ Sec t =◦ Cot t =

Therefore, the number t on the number line always wraps onto the point (cos t, sin t) on the unit circle

P(cos t, sin t)

Exploring the Unit CircleExploring the Unit Circle1. For any t, the value of cos t lies between -1 and 1 inclusive.

2. For any t, the value of sin t lies between -1 and 1 inclusive.

3. The values of cos t and cos (-t) are always opposites of each other. (Recall that this is the check for an even function.)

4. The values of sin t and sin(-t) are always opposites of each other. (Recall that this is the check for an odd function.)

5. The values of sin t and sin (t +2) are always equal to each other. In fact, that is true of all six trig functions on their domains, and for the same reason.

6. The values of sin t and sin (t+) are always opposites of each other. The same is true of cost and cos (t+)

7. The values of tan t and tan (t+) are always equal to each other (unless they are both undefined).

8. The sum (cost)2 + (sint)2 always equal 1.

Periodic FunctionsPeriodic FunctionsDef: Periodic Functions

◦A function y=f(t) is periodic if there is a positive number c such that f(t+c) = f(t) for all values of t in the domain of f. The smallest such number c is called the period of the function.

Using PeriodicityUsing PeriodicitySin (57801/2)

Cont.. Cont.. Cos(288.45) – cos(280.45)

Tan(/4 – 99,999)

16-Point Unit Circle16-Point Unit Circle

The 16- Point Unit CircleThe 16- Point Unit Circle

4.4 – Graphs of Sine and 4.4 – Graphs of Sine and Cosine: SinusoidsCosine: Sinusoids

HW: Pg. 393-394 #1-16e, 50-56e

Graph of:Graph of:

F(x)=sinx F(x)=cosx

Sinusoids and Sinusoids and TransformationsTransformationsA function is a sinusoid if it can

be written in the form

◦F(x) = a sin(bx+c)+d

Where a, b, c, and d are constants and a,b≠0.

TransformationsTransformationsHorizontal Stretches and Shrinks

affect the period and the frequency

Vertical Stretches and Shrink affect the amplitude

Horizontal translations bring about phase shifts

Amplitude of a SinusoidAmplitude of a SinusoidThe amplitude of the sinusoid

f(x)=asin(bx+c)+d is |a|

F(x)=acos(bx+c)+d

◦Graphically, the amplitude is half the height of the wave

Vertical Stretch or Shrink Vertical Stretch or Shrink and Amplitudeand AmplitudeFind the amplitude of each

function and describe how the graphs are related◦Y1=cosx

◦Y2= 1/2cosx

◦Y3 = -3cosx

Period of a SinusoidPeriod of a SinusoidThe period of the sinusoid

f(x)=asin(bx+c)+d is 2/|b|.

◦Graphically, the period is the length of one full cycle of the wave.

Find the period of each Find the period of each function and describe how function and describe how the graphs are related:the graphs are related:Y1= sinx

Y2= -2sin(x/3)

Y3= 3sin(-2x)

Frequency of a SinusoidFrequency of a SinusoidThe frequency of the sinusoid

f(x)=asin(bx+c)+d is |b|/2

◦Graphically, the frequency is the number of complete cycles the wave completes in a unit interval.

Finding the Frequency of a Finding the Frequency of a SinusoidSinusoidFind the frequency of the

function f(x)=4sin(2x/3). What does it mean graphically?

Phase ShiftPhase ShiftWrite the cosine function as a

phase shift of the sine function.

Write the sine function as a phase shift of the cosine function.

Construct a sinusoid with Construct a sinusoid with period /5 and amplitude 6 that period /5 and amplitude 6 that goes through (2,0)goes through (2,0)

Graphs of SinusoidsGraphs of SinusoidsThe graphs of y=asin(b(x-h))+k and

y=acos(b(x-h))+k have the following characteristics:◦ Amplitude=|a|◦ Period=2/|b|◦ Frequency= |b|/2

Where compared to the graphs of y=asinbx and y=acosbx, they also have the following characteristics: A phase shift of h A vertical translation of k

Construct a sinusoid y=f(x) Construct a sinusoid y=f(x) that rises from a minimum that rises from a minimum value of y=5 at x=0 to a value of y=5 at x=0 to a maximum of y=25 at x=32.maximum of y=25 at x=32.

4.5-Graphs of Tangent, 4.5-Graphs of Tangent, Cotangent, Secant, and Cotangent, Secant, and CosecantCosecant

HW: Pg. 402 #1-20e

The Tangent FunctionThe Tangent FunctionF(x)=tanxDomain:Range:Continuous? Inc? Dec?Symmetry?Bounded?Extrema?H.A?V.A?End Behavior:

Tanx = sinx/cosx

◦Y=a tan(b(x – h)) + k

Graphing a Tangent Graphing a Tangent FunctionFunctiony=-tan2x

◦Hint: Find Vertical Asymptotes and graph four periods of the function.

The Cotangent FunctionThe Cotangent FunctionThe cotangent function is the

reciprocal of the tangent function:◦Cot x = cosx/sinx

Describe the graph of Describe the graph of f(x)=3cot(x/2) +1.f(x)=3cot(x/2) +1.Locate the vertical asymptotes

and graph two periods

The Secant FunctionThe Secant FunctionReciprocal of the cosine function

secx = ◦Work with partner to answer:

Whenever cosx=1, what does secx equal?

When does the secant function have asymptotes?

What is the period of secx ?

Compare extrema with cosx and secx.

Quiz (4.1-4.5) :Quiz (4.1-4.5) :

Solving a Trigonometric Solving a Trigonometric Equation AlgebraicallyEquation AlgebraicallyFind the value of x between

and 3/2 that solves secx= -2

The Cosecant FunctionThe Cosecant FunctionCscx =

◦Whenever sinx = 1, what does cscx equal?

◦ Where are the asymptotes of the graph of the cosecant function?

◦What is the period of cscx?◦What do you notice about the

extrema of y=sinx and y=cscx?

Solving a Trig Equation Solving a Trig Equation GraphicallyGraphicallyFind the smallest positive

number x such that x2 =cscx

State the period and State the period and frequency of the functions:frequency of the functions:Y=cos2xY=sin1/3xY=sin3xY=cos1/2x

State the sign (positive or State the sign (positive or negative) of the sine, cosine, negative) of the sine, cosine, and tangent in the quadrant:and tangent in the quadrant:1. Quadrant I

2. Quadrant II

3. Quadrant III

4. Quadrant IV

Find exact value Find exact value algebraicallyalgebraically::Sin(/6)Cos(2/3)Sin(-/6)Tan(/4)Sin(2/3)Cos(-/3)

Do Now (1/6):

1. Find the period, amplitude, frequency and phase shift of the function:◦Y=-3/2sin2x

2. Construct a sinusoid with the given amplitude and period that goes through the given point: Amplitude:3, period:π, point:(0,0)

4.6 - Graphs of Composite Trigonometric Functions

HW: Pg. 411-412 #9-28e, 39-42

Combining Trigonometric and Algebraic FunctionsWhen combining sine function

with x2, which of the following functions appear to be periodic for -2π≤x≤2π:◦Y=sinx + x2

◦Y=x2sinx◦Y=(sinx)2

◦Y=sin(x2)

Sinx and x2 Combinations:

Verifying Periodicity algebraically:Verify that f(x)=(sinx)2 is periodic

and determine its period graphically.

Prove algebraically that f(x)=sin3x is periodic and find the period graphically. State the domain and

range.

Analyzing nonnegative periodic functionsFind the domain, range, and

period of each of the following functions. Sketch a graph showing four periods.◦F(x)=|tanx|

◦G(x)=|sinx|

Adding a Sinusoid to a Linear FunctionThe graph of f(x)=0.5x+sinx

oscillates between two parallel lines. What are the equations of the two lines?

Investigating Sinusoids:Graph these functions, one at a time, in

the viewing window [-2π,2π] by [-6,6]. Which ones appear to be sinusoids?◦ Y=3sinx + 2cosx◦ Y=2sin3x - 4cos2x ◦ Y=cos((7x-2)/5) + sin(7x/5)◦ Y=2sinx - 3cosx◦ Y=2sin(5x+1) - 5cos5x◦ Y=3cos2x + 2sin7x

What relationship between the sine and cosine functions ensures that their sum or difference will again be a sinusoid? Check your guess on a graphing calculator by constructing your own examples.

The rule is simple: Sums and differences of sinusoids with the same period are sinusoids.Sums That Are Sinusoid

Functions◦If y1=a1sin(b(x-h1)) and y2=a2cos(b(x-

h2)), then

y1+y2 = a1sin(b(x-h1)) + a2(cos(b(x-h2))

is a sinusoid with period 2π/|b|

Determine whether the following functions are sinusoids:

F(x)=5cosx +3sinxF(x)=cos5x + sin3xF(x)=2cos3x-3cos2xF(x)=acos(3x/7)-bcos(3x/

7)+csin(3x/7)

Expressing the sum of sinusoids as a Sinusoid:Let f(x)=2sinx+5cosx

(a) Find the period of f(b) Estimate the amplitude and phase shift graphically(to the nearest hundredth)(c) Give a sinusoid asin(b(x-h)) that approximates f(x).

Showing a function is periodic but not a sinusoidF(x)=sin2x + cos3x

We need to show that f(x+2π)=f(x)

Damped Oscillation

F(x)=(x2+5)cos6x

The “squeezing” effect is called ______________.

Damped OscillationThe graph of y=f(x)cosbx (or

y=f(x)sinbx) oscillates between the graphs of y=f(x) and y=-f(x). When this reduces the amplitude of the wave, it is called damped oscillation. The factor f(x) is called the damping factor.

Identify damped oscillation:F(x)=3-xsin4xF(x)=4cos5xF(x)=-2xcosx

Do Now :Do Now :Find the two parallel lines the

following functions are oscillating between:

◦Y=2x + cosx

◦Y=2-0.3x+cosx

4.7 – Inverse 4.7 – Inverse Trigonometric FunctionsTrigonometric Functions

HW: Pg. 421 #1-12e, 24-28e

Solve for x :Solve for x :Sinx = ½

Cosx= 1

Sin x = √(2)/2

Inverse Sine FunctionInverse Sine Functionx=siny y=sin-1x

Inverse Sine Function Inverse Sine Function (Arcsine Function)(Arcsine Function)The unique angle y in the interval

[-2π,2π] such that siny=x is the inverse sine (or arcsine) of x, denoted sin-1x or arcsinx.

The domain of y=sin-1x is ____ and the range is_______

Y=sinY=sin-1-1xxAlong the right-hand side of the

unit circle

Evaluating sinEvaluating sin-1-1x without a x without a calculatorcalculatorFind the exact value of each

expression without a calculator.(a)sin-1(1/2)

(b)sin-1(-√(3)/2)

(c)sin-1(π/2)

(d)sin-1(sin(π /9))

(e)sin-1(sin(5 π /6))

Now, with a calculator:Now, with a calculator:sin-1(-.91)

sin-1(sin(3.49π))

Inverse CosineInverse Cosine

Inverse Cosine Function Inverse Cosine Function (Arccosine function)(Arccosine function)

The unique angle y in the interval [0,π] such that cosy=x is the inverse cosine (or arccosine) of x, denoted cos-1x or arccosx.

The domain of y= cos-1x is ____ and the range is ______.

Inverse TangentInverse Tangent

Inverse Tangent Function Inverse Tangent Function (Arctangent Function)(Arctangent Function)The unique angle y in the interval

(-π/2,π/2) such that tany=x is the inverse tangent (or arctangent) of x, denoted y=tan-1x or arctanx.

The domain of y=tan-1x is _____ and the range is ______.

Evaluate without Evaluate without calculator:calculator:(a) y=cos-1(-√(2)/2)(b)y=tan-1(√3)(c) y=cos-1(cos(-1.1))

Describe the end behavior Describe the end behavior of:of:

y=tan-1x

Exploration: Finding Inverse Exploration: Finding Inverse Trig Functions of Trig Trig Functions of Trig FunctionsFunctionsIn the right triangle shown, the

angle is measured in radians

1. Find tan

2. Find tan-1x

3. Find the hypotenuse of the triangle as a function of x.

4. Find sin(tan-1x) as a ratio involving no trig functions.

5. Find sec(tan-1x) as a ratio involving no trig functions.

6. If x<0, then y=tan-1x is a negative angle in the IV Quadrant. Verify that your answers to parts (4) and (5) are still valid in this case.

x

1

Composing trig functions with ArcsineCompose each of the six basic

trig functions with sin-1x.

4.8 - Solving Problems with Trigonometry

HW: Pg.431-433 #4-10e, 16, 22

DEF:Angle of elevation - the angle

made when the eye moves up from horizontal to look at something above.

Angle of depression - the angle made when the eye moves down from horizontal to look at something below.

Using angle of depressionThe angle of depression of a buoy from the

top of the Barnegat Bay lighthouse 130 feet above the surface of the water is 6º. Find the distance x from the base of the lighthouse to the buoy.

Making Indirect MeasurementsFrom the top of the 100-ft-tall Weehawken

Tower, a man observes a car moving toward the building. If the angle of depression of the car changes from 22 to 46 during the period of observation, how far does the car travel?

Finding Height above Ground A large, helium filled Football is moored at the

beginning of a parade route awaiting the start of the parade. Two cables attached to the underside of the football make angles of 48º and 40º with the ground and are in the same plane as a perpendicular line from the football to the ground. If the cables are attached to the ground 10 ft from each other, how high above the ground is the football?

Using Trigonometry in NavigationA US Coast Guard patrol boat leaves Port

Cleveland and averages 35 knots (nautical mph) traveling for 2 hrs on a course of 53º and then 3 hrs on a course of 143º. What is the boat’s bearing and distance from Port Cleveland?

Practice:Practice:Pg 431 #1-3