Chapter 4 – Trigonometric Functions

4.1 – Angles and Their Measures Degrees and Radians Degree – 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 measure (a) Convert 37.425 ̊ to DMS (b) Convert 42 ̊24’36’’ to degrees.

Radian A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

Working with Radian Measure How many radians are in 90 degrees? How many degrees are in /3 radians?

Cont… Find the length of an arc intercepted by a central angle of ½ radian in a circle of radius 5 inches.

Degree-Radian Conversion To convert radians to degrees, multiply by To convert degrees to radians, multiply by

Practice: Work with partners Pg 356 # 1-24e

Do Now: Convert from radians to degrees: π/6 π/10 5π/9

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

Solving a Right Triangle A 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 the base of the Tower of Weehawken, the angle of elevation 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 Angles Two angles can have the same initial side and the same terminal side, yet have different measures - called coterminal angles.

Find Coterminal Angles 40 ̊ -160 ̊ 2/3 radians

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

Evaluating Trig Functions Determined by a Point in QI Let θ be the acute angle in standard position whose terminal side contains the point (4,3). Find the six trigonometric functions of θ.

Evaluating Trig Functions Determined by a Point NOT in QI Let θ be an angle in standard position whose terminal side contains the point (-5,2). Find the six trigonometric functions of θ.

Trigonometric Functions of any Angle Let 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 r θ x

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

Evaluating More Trig Functions Sin(-210 ̊) Tan(5/3) Sec(-3/4)

… Sin(-270 ̊) Tan(3) Sec(11/2)

When are sin, cos, and tan positive? Quadrant 1- Quadrant II – Quadrant III – Quadrant IV –

Using One Ratio to Find the Others Find cos θ and tan θ 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 own Find 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 Real Numbers DEF: Unit Circle The unit circle is a circle of radius 1 centered at the origin.

Wrapping Function Connects points on a number line with points on the circle

Trigonometric Functions of Real Numbers Let 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 Circle For any t, the value of cos t lies between -1 and 1 inclusive. For any t, the value of sin t lies between -1 and 1 inclusive. The values of cos t and cos (-t) are always opposites of each other. (Recall that this is the check for an even function.) The values of sin t and sin(-t) are always opposites of each other. (Recall that this is the check for an odd function.) 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. The values of sin t and sin (t+) are always opposites of each other. The same is true of cost and cos (t+) The values of tan t and tan (t+) are always equal to each other (unless they are both undefined). The sum (cost)2 + (sint)2 always equal 1.

Periodic Functions Def: 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 Periodicity Sin (57801/2)

Cont.. Cos(288.45) – cos(280.45) Tan(/4 – 99,999)

16-Point Unit Circle

The 16- Point Unit Circle

4.4 – Graphs of Sine and Cosine: Sinusoids HW: Pg. 393-394 #1-16e, 50-56e

Graph of: F(x)=sinx F(x)=cosx

Sinusoids and Transformations A 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.

Transformations Horizontal 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 Sinusoid The 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 and Amplitude Find the amplitude of each function and describe how the graphs are related Y1=cosx Y2= 1/2cosx Y3 = -3cosx

Period of a Sinusoid The 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 function and describe how the graphs are related: Y1= sinx Y2= -2sin(x/3) Y3= 3sin(-2x)

Frequency of a Sinusoid The 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 Sinusoid Find the frequency of the function f(x)=4sin(2x/3). What does it mean graphically?

Phase Shift Write 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 period /5 and amplitude 6 that goes through (2,0)

Graphs of Sinusoids The 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) that rises from a minimum value of y=5 at x=0 to a maximum of y=25 at x=32.

4.5-Graphs of Tangent, Cotangent, Secant, and Cosecant HW: Pg. 402 #1-20e

The Tangent Function F(x)=tanx Domain: 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 Function y=-tan2x Hint: Find Vertical Asymptotes and graph four periods of the function.

The Cotangent Function The cotangent function is the reciprocal of the tangent function: Cot x = cosx/sinx

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

The Secant Function Reciprocal 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) : Convert from DMS-Degrees Convert Degree-Radians Evaluate Trigonometric Functions Use Trigonometric Ratio to find the others Solving a Right Triangle Find Coterminal Angles Evaluate Trig Functions Determined by a Point in different Quadrants Evaluate Trig Functions of angles (in degrees or radians) Use Periodicity Find amplitude, period, and frequency of sinusoids Describe how trig functions are transformed Locate vertical asymptotes and periods of all trig functions

Solving a Trigonometric Equation Algebraically Find the value of x between  and 3/2 that solves secx= -2

The Cosecant Function Cscx = 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 Graphically Find the smallest positive number x such that x2 =cscx

State the period and frequency of the functions: Y=cos2x Y=sin1/3x Y=sin3x Y=cos1/2x

State the sign (positive or negative) of the sine, cosine, and tangent in the quadrant: 1. Quadrant I 2. Quadrant II 3. Quadrant III 4. Quadrant IV

Find exact value algebraically: 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 Functions When 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 functions Find 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 Function The 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.

Sums That Are Sinusoid Functions 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 +3sinx F(x)=cos5x + sin3x F(x)=2cos3x-3cos2x F(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 sinusoid F(x)=sin2x + cos3x We need to show that f(x+2π)=f(x)

The “squeezing” effect is called ______________. Damped Oscillation F(x)=(x2+5)cos6x The “squeezing” effect is called ______________.

Damped Oscillation The 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-xsin4x F(x)=4cos5x F(x)=-2xcosx

Do Now : Find the two parallel lines the following functions are oscillating between: Y=2x + cosx Y=2-0.3x+cosx

4.7 – Inverse Trigonometric Functions HW: Pg. 421 #1-12e, 24-28e

Solve for x : Sinx = ½ Cosx= 1 Sin x = √(2)/2

Inverse Sine Function x=siny y=sin-1x

Inverse Sine 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=sin-1x Along the right-hand side of the unit circle

Evaluating sin-1x without a calculator Find the exact value of each expression without a calculator. sin-1(1/2) sin-1(-√(3)/2) sin-1(π/2) sin-1(sin(π /9)) sin-1(sin(5 π /6))

Now, with a calculator: sin-1(-.91) sin-1(sin(3.49π))

Inverse Cosine

Inverse Cosine 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 Tangent

Inverse Tangent 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 calculator: y=cos-1(-√(2)/2) y=tan-1(√3) y=cos-1(cos(-1.1))

Describe the end behavior of: y=tan-1x

Exploration: Finding Inverse Trig Functions of Trig Functions In 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 Arcsine Compose 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 depression The 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 Measurements From 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 Navigation A 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: Pg 431 #1-3