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Chapter 4 – Trigonometric Functions. 4.1 – Angles and Their Measures Degrees and Radians. Degree – denoted ̊ , is a unit of angular measure equal to 1/180 th of a straight angle.
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4.1 – Angles and Their MeasuresDegrees 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=rθ • 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=πrθ/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= sinθ = • Cosineθ = cosθ = • Tangentθ = tanθ = • Cosecantθ = cscθ = hyp/opp • Secantθ = secθ = hyp/adj • Cotangentθ = cotθ = 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. 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: • 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. y P(x,y) r θ x
Evaluating Trig Functions Determined by a Point in QI • Letθ 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 • Letθ 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= • cosθ= • tanθ= • cscθ= • secθ= • cotθ= 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 • secθ=3 and sinθ>0 • cotθ is undefined and secθ is negative
…Try on your own • Find sinθ and tanθ if cosθ=2/3 and cotθ>0 • Find tanθ and secθ if sinθ=-2/5 and cosθ>0 • Find secθ and cscθ if tan θ=-4/3 and sinθ>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)
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)