I. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.

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1 0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:. 60 minutes equal degree. 60 seconds equal minute. Also, 600 seconds equal degree C. Eample : working with DMS measure. Approimate the angle o 0 as a decimal to the nearest 0,000 th of a degree. 0 '0" Approimate the angle.9 o in terms of degrees, minutes, and seconds ' + ' 60 '8." D. In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north. W - N S E E. Radians. A central angle of a circle has measure radian if it intercepts an arc with the same length as the radius. a radian. Conversion between degrees and radians 60 radians radians 80 a. Eample : a. How man radians are in 7 degrees? 7 radians 80 b. How man degrees are in Radians? 80 6 c. Find the length of an arc intercepted b a central angle of radians in a circle of radius 6 inches. radian intercepts radius, or 6 inches a central angle of radians intercepts 8 inches.

2 0//0 II. Circular Arc Length A. Arc length formulas: if theta is a central angle in a circle of radius r, the following formulas will give the arc length s, Radians: s r r Degrees: s 80 B. Eample :. Suppose that an angle inscribed within a circle of radius centimeters has measure. radians. Determine the length of the arc defined b the angle. s. 9.cm. Eample : Suppose that a central angle of a circle of radius meters intercepts an arc of length meters. Find the radian measure of the angle. 7 6 radians C. Eample : A person is seated on the end of a see saw whose total length is m. The see saw moves up and down through a 8 degree angle ever seconds. Through what distance does the person move in a minute?. 6 s.m 80.m 8.86m sec 60sec 8.86meters III. Angular and Linear Motion A. We can measure speed in both linear (such as miles per hour) and angular speed (such as revolutions per minute). B. Eample : Suppose that the wheels on a tractor have a diameter of 0 inches and that the angular speed of the tires is 0 rotations per minute. What is the truck s speed in miles per hour? 0rot 60 min radians ft 0in mi min hr rot in radian 80 ft 6. mi hr C. Nautical Mile. A nautical mile is the length of minute of arc (/60 of a degree) length along the earth s equator.. The familiar land mile is called the statute mile.. We find the conversion below (the radius of the earth is approimatel 96 stat miles): rad ' ,800 radians naut mile 96 stat mi 0,800. stat mi 0,800 stat mi naut mi 0.87 naut mi 96 distance conversions stat mile 0.87 naut mile naut mile. statute mile. Eample 6: A pilot flies from New Jerse to Florida, a distance of approimatel 9 miles. How man nautical miles is it from New Jerse to Florida. 9 0,800 9 stat mi 89 naut mi. 96

3 0//0 I. Right Triangle Trigonometr A. Definition: Trigonometric Functions Let theta be an acute angle in the right triangle ABC, Then: B Hpoteuse Opposite A C Adjacent opp adj opp sin cos tan hp hp adj hp hp adj csc sec cot opp adj opp B. Standard Position An angle is in standard position if it has one ra along the positive -ais. II. Two Famous Triangles A. Eample : find the values of all si trigonometric functions for an acute angle of degrees. opp. sin hp adj. cos hp opp. tan adj. csc opp.... sec adj... adj 6. cot opp B. Eample : find the values of all si trigonometric functions for an acute angle of 0 degrees. C. Eample : find the values of all si trigonometric functions for an acute angle of 60 degrees. opp. sin0 hp. cos0 adj hp opp. tan adj hp adj. csc0 6. cot opp opp hp. sec0... adj 0 opp. sin hp adj. cos60 hp opp. tan adj 60 hp. csc60... opp hp adj. sec60 6. cot adj opp

4 0//0 D. Eample : Let theta be an acute angle such that sin(theta)/. Evaluate the other five trigonometric functions of theta. 9 cos tan csc sec... cot.8... B III. Applications of Right Triangle Trigonometr A. Solve the following right triangle (find the measures of all the lengths and all the angles). 0.6 a cos a b sin.9 C b 67 cos a 0.6 A sin b.98 h B. A surveor wishes to determine the height of a building. She measures a distance of 00 feet from the center of the base of the building and uses a transit to determine the angle of inclination (elevation) to the top of the building. This angle is o 7. opp tan ( 7' ) adj h tan ( 7' ) tan 7' h ft h 7' 00 ft

5 0//0 I. Trigonometric Functions of An Angle.: Trigonometr Etended: The Circular Functions Terminal Side measure of the angle verte α Initial Side Two Angles in Standard Position α A positive angle (counterclockwise) Coterminal Angles α β - Positive and negative coterminal angles β A negative angle (clockwise) α β Two positive coterminal angles A. Eample : Finding coterminal angles. Find a positive and negative coterminal angle for each of the following angles B. Eample : Evaluating trig functions determined b a point. - Let theta be the acute angle in standard position whose terminal side contains the point (,-7). Find the si - trigonometric functions of theta the distance from the point to the origin is 8 (,-7) sin cos tan csc sec 8.9 cot 7

6 0//0 P(,) - C. Definition: Trigonometric Functions of an angle. Let theta be an angle in standard position, and let P(,) be an point on the terminal side of the angle (ecept for the origin). Let r denote the distance from P to the origin. Then: r r sin csc, 0 r r cos sec, 0 r tan, 0 cot, 0 E. Eample : Evaluating a trig function using a reference triangle. Find the si trig functions at an angle of 0 degrees. Notice that this forms the triangle below (called the reference triangle) sin csc cos sec tan cot F. Eample : Find the following without a calculator: ( ). sin 7. csc G. Angles whose terminal sides lie along one of the coordinate aes are called quadrantal angles, and although the do not produce reference triangles at all, so we use the coordinates on the unit circle instead. H. Eample : Evaluating trig functions of quadrantal angles.. cos this lies on the positive -ais, thus the coordinates are (0,) 0 cos 0 r. csc 900 this lies on the negative -ais, thus the coordinates are (-,0) csc 900 r, which is undefined 0

7 0//0 I. Eample : Find the cosine and tangent of theta, given the following information.. sin and cos < 0 second quadrant, since sin>0 and cos<0 P cos tan. sec and cot < 0 7 cos tan P. csc is undefined and sec < 0 This is on the negative -ais cos tan 0 II. Trigonometric Functions of Real Numbers A. Definition: Unit Circle The unit circle is a circle of radius centered at the origin P(cos(t),sin(t)) ( ) C. Definition: Trigonometric Functions of Real Numbers. Let t be an real number, and let P(,) be the point corresponding to t when the number line is wrapped onto the unit circle. C 0. - B t t sin cos csc, 0 sec, 0 tan, 0 cot, III. Periodic Functions A. Definition: Periodic Function A function f() is periodic if there is a positive number c such that (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. B. Eample : Using Periodicit. Find each of the following numbers without a calculator cos cos + cos + cos 0 IV. The 6-Point Unit Circle ( ) ( ) sin( ) sin ( 89.). sin 8. sin 89. since these wrap to the same point tan,7 tan

8 0//0.: Graphs of Sine and Cosine: Sinusoids I. Sinusoids and Transformations A. Both sine and cosine graphs are considered sinusoids (cosine graphs are sine graphs under a translation) B. Definition: A function is a sinusoid if it can be written in the form: f asin( b+ c) + d Where a, b, c, and d are constants, and neither a nor b can be zero. C. Definition: Amplitude of a Sinusoid The amplitude of a sinusoid is a Graphicall, it is half the height of the wave. D. Eample : find the amplitude of each function, and describe how these are related.. f cos. f ( ) cos amp amp vertical stretch of. f cos amp vertical stretch of, reflected over -ais E. General forms of sinusoids: F. Period of a Sinusoid. f asin b+ c + d f acos b+ c + d The period of a sinusoid is b Graphicall, it is the length of one full ccle of the wave. G. Eample : Determine the periods of the following functions.. sin ( ). sin ( ) b b. sin. cos 6 6 b ( ) b ( ) II. Frequenc of a sinusoid A. Frequenc is the number of complete ccles of the function in a given interval. The frequenc of sinusoids is below: b frequenc this is the reciprocal of the period

9 0//0 B. Eample : Find the frequenc of the function below, and interpret the meaning graphicall. f cos() f cos frequenc, 6 which is the reciprocal of the period of The function goes through one ccle ever units I. Eample : Write the cosine function as a phase shift of the sine function. g cos f sin - We see that the sine function must be shifted pi/ units to the left to get the cosine curve, thus: cos sin ( + ) similarl: sin cos period: b J. Eample : Construct a sinusoid with period and amplitude that goes through (,0). b± 8, we arbitraril use b 8 amplitude: a a ±,we use a we now have f sin 8 in order to shift to the point (,0): f sin 8 sin 8 K. Summar for Sinusoids: The graphs of the following sinusoids (where a and b are both nonzero) have the following characteristics: amplitude a period b frequenc b asin b h + k acos b h + k phase shift h vertical shift k L. Eample 6: Construct a sinusoid with the following characteristics: minimum (0, ) maimum (, ) ( ) amplitude period 8 8 b ± b rather than shift a sine curve, we can just reflect cosine over the -ais to have a minimum at 0 cos ± cos finall, we must verticall shift the graph [ ] [ ] we have range, we want range, k 9 cos + 9

10 0//0 II. Modeling Periodic Behavior with Sinusoids. A. Steps:. Determine the Maimum value M and minimum value m. The amplitude A of the sinusoid and vertical shift C will be: M m ; M + A C m. Determine the period p, the time interval of a single ccle of the period function. The horizontal shrink (or stretch) will be: B p. Choose an appropriate sinusoid based on behavior at some given time T. f( t) Acos B t T + C attains a maimum value f( t) Acos B t T + C attains a minimum value f ( t) Asin B t T + C is halfwa between a minimum and a maimum value f( t) Asin B t T + C is halfwa between a maimum and a minimum value B. Eample 7: A student is doing pushups, one complete push-up takes seconds. The student starts the push-up at 0 inches off of the floor, and reaches a minimum of inches from the floor.. Find a sinusoidal function that can model this situation. M m 0 Amp: A 8. Pd sec B B M + m 0 + V. Shift: C. since it starts at a peak, we use Cosine and T 0 f( t) 8.cos t What was the height after 8. seconds? f (8.) 8.cos ( 8.) in.. After how man seconds does the student reach a height of 0 inches from the floor for the first time? (0.8sec,0 in) g 0 f 8. cos( ( ) )

11 0//0 I. The Tangent Function A. Remember that the tangent is defined as follows: sin tan cos - - B. Note that the tangent function has asmptotes where the cosine is zero. C. The tangent graph has zeros wherever the sine function is zero. f()tan() f()tan() h cos q sin - - D. Eample : Sketch the graph of the function below. f tan First, shift the graph units to the right. Now, scale horizontall b a factor of / r tan - ( ) s tan - ( ) f()tan() r tan - ( )

12 0//0 tan(-pi/) This has been rescaled, and plotted with the original tangent graph II. The Cotangent Function A. The cotangent function is defined as follows: cos cot sin t tan B. Note that the cotangent function has asmptotes where the sine is zero. C. The cotangent graph has zeros wherever the cosine function is zero. t sin u cos III. The Secant Function sec cos g cos O O secant has asmptotes where cosine is zero the graphs share the values at the 'top' of the unit circle, where cosine or -

13 0//0 IV. The Cosecant Function t sin cosecant has asmptotes where sine is zero the graphs share the values at the 'top' of the unit circle, where sine or -

14 0//0 I. Combining Trigonometric and Algebraic Functions. A. Eample : Graph each of the following. Which of the functions appear to be periodic?. g cos. g cos f cos + f cos g g h cos h cos. f cos. k cos q cos( ) g g s cos - h cos - - r cos - wh is this the same as the cosine graph?

15 0//0 B. Eample : Verif that k cos is periodic and determine its period. ( ) k ( + ) cos + cos k this means the function is periodic, but there ma be a smaller number for the period we see from the graph that the period is C. Eample : Find the domain, range, and period of the following function. f cot The domain of the function will be the same as the domain of the cotangent function: dom R, n, n 0, ±, ±,... rng period [ 0, ) 8 6 f tan D. Eample : The graph of f + cos oscillates between two parallel lines. What are the equations of the two lines? The two lines will have the same slope as the line f()/, but will have -intercepts one above and one below (because cosine oscillates between - and ) g + h - f +cos - II. Sums and differences of sinusoids. A. Sums and differences of sinusoids with the same period are again sinusoids (unless the are eact opposites, and thus cancel out to a horizontal line). sin If a sin b h and asin b h, + asin b h + a b h is a sinusoid with period b B. Eample : Determine which of the following functions is a sinusoid.. f sin sin no. f cos sin( ) es. f cos sin no. f sin sin + + 7cos es

16 0//0 C. Eample 6: Epress the following function as a single sinusoid (graphicall) f () cos( )- sin( ) ma ( ) f.606sin etension We could also use the following cosine function: ( ) f ( ).606cos int.079 since the function has no vertical shift, this is the phase shift Period,so b (like both parts of the sum) B how much do the phase shifts differ? Note: This is 0.78, wh not? D. Eample 7: calculate the period of the following functions (which are not sinusoids). This period will be the least common multiple of the periods of each function.. f cos sin. f sin + sin + sin periods,, LCM period periods, LCM period III. Damped Oscillation The graph of fcos (or fsin ) oscillates between the graphs of f and f. This is called a damped oscillation. The factor f ( ) is called the damping factor.. cos sin h e g e h q () cos q e -0. sin f sin -0-0 r cos f h -e-0. -

17 0//0. f ln cos g ln f cos q ln cos 0 - h -ln

18 0//0 I. Inverse Sine Function Arcsin A.) We have to restrict the domain of f () sin to In general: Domain: [,], Range:, sin B.) E. Find the eact values of each without a calculator. sin.) sin.) sin 6.) sin sin 7.) sin sin 7 II. Inverse Cosine Function Arccos A.) We have to restrict the domain of f () cos to [ ] 0, cos In general: Domain: Range: [,] [ 0, ] III. Inverse cos Tangent Function Arctan A.) We have to restrict the domain of f () tan to In general: Domain: (, ), Range: B.) E. Find the eact values of each without a calculator..) cos.) tan (.) tan sin ) IV. Composing Trig and Arc Trig Functions sin sin A.) ALWAYS TRUE cos( cos ) tan ( tan ) B.) ONLY TRUE ON RESTRICTED DOMAINS sin cos tan ( sin ) ( cos ) ( tan ) C.) Given the triangle as shown with measured in radians, find the following..) tan.) tan.) The Hpotenuse.) sin tan.) sec tan sin + + sec +

19 0//0 C.) E.- Compose each of the si basic trig functions cos with and reduce the composite function to an algebraic epression involving no trig functions. First, we need to draw the triangle where cos ( ) sin cos tan cos sec cos cos cos cot cos csc cos

20 0//0 I. Angles of depression and elevation angle of depression.8: Solving Problems With Trigonometr. angle of elevation Eample : In order for paratroopers to land on a target when jumping from a C-7 traveling at mph at an altitude of 00 ft, the jump when the angle of depression to the target is o 0. a. How far from the target horizontall will the be when the jump? ft tan ( 0' ) tan ( 0' ) b. How far will the travel in the air? (assume the move in a straight line) plane 00 0' sin( 0' ) ft 9ft sin 0' 0' target II. Simple Harmonic Motion A. A point moving on a number line is in simple harmonic motion if its directed distance d from the origin is given b either d asin ωt or d acosωt ω. a and (omega) are real numbers. ω > 0. The motion has the frequenc below, which is the number of oscillations per unit of time. ω f B. Eample : A block is attached to a hanging spring and set in motion b compressing the spring from its rest position and releasing it. The angular velocit of the spring s motion is 6pi radians/sec, and the amplitude of the oscillation is 7 inches.. Write an equation for the motion of the spring. () 7cos( 6 t) d t. What is the frequenc of the spring? 6 f rev/sec. How far from the starting point is the spring eactl.6 seconds after it s released. ( ) d.6 7 cos 6.6.6in in C. A weight is attached to a spring and set in motion b compressing the spring from its rest position and releasing it. The maimum displacement of the spring is cm, and it takes second to complete one ccle.. Write an equation for the motion of the spring. amp ω ω starts at a maimum height, so we use cosine dt () cos ( t)

21 0//0. What is the displacement of the spring after. seconds. ( ) d(.) cos..6in. What is the total distance traveled b the spring during the first. seconds. during the first seconds, it travels up and down times: (6) 8in during the last 0. seconds, it travels:.6.76in distance in Sketch a graph of the function on the domain [0,6] f cos( ) 6