1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint of the ray. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis. Positive angle a counterclockwise rotation Negative angle- a clockwise rotation. II. Degree 1 revolution = 36 1 revolution 18 2 1 revolution 9 4 1 revolution 1 36
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3 Coterminal Angles If angles α and β have the same initial and terminal sides, then they are coterminal and Example 36 n, where n {, -2, -1,, 1, 2, } a) Find a angles coterminal to 39. b) Find angles coterminal to -12 Complimentary - 9 Supplementary - 18 α and β must be POSITIVE ANGLES Example: Find the compliment and the supplement of a) θ = 5 b) θ = 148
4 III. Degrees Minutes Seconds Notation D M ' S " 1 = one minute = 1 6 => 1 6 ' 6 ' or 1 1 = one second = 1 36 => 1 36" or Example: Write 64 degrees, 32 minutes, 47 seconds in D M ' S " notation. Example: Convert 64 32'47" to decimal form. 1. Place values before in front of the decimal 2. Multiply the minutes by 1 6 ' 3. Multiply the seconds by 4. Add the values. 1 36" Example: Convert 18.255 to D M ' S " form. 1. Write the part in front of the decimal without the 6 ' 2. Multiply the decimal by. Attach. 1 3. If the decimal still exists, multiply that decimal by 6" ' 1. Attach.
5 IV. Radians Measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. The Arclength (s) = radius (r) when θ = 1 radian, so: 1revolution 2 radian 1 revolution radian 2 1 revolution radian 4 2
6 Example: Find the compliment of 12 Example: Find the supplement of 5 6 Example: Find a coterminal angle to 17 6. Generally, α is coterminal with 2 n, n {, -2, -1,, 1, 2, }.
7 V. Degrees and Radians conversions radians 1. Degrees to radians: multiply the degrees by 18 2. Radians to degrees: multiply the radians by 18 radians Example: Convert 135 to radians. Find a negative coterminal angle. Example: Convert 2 radians to degrees. Find a positive coterminal angle.
8 VI. Arc Length: s r, where θ is measured in RADIANS. What is the distance How far does the tip move? Example: Find the arc length when r = 15 inches and θ = 6. Example: Find the radian measure of the central angle for the circle below.
9 Example: What is the difference in latitudes of Charleston, WV ( 38 21'N )and Jacksonville, Fl ( 3 2' N )? Assume that the cities lie along the same longitudinal line, and that the radius of the Earth is 396 miles. 1. Convert each distance to decimal form (degrees). 2. Subtract distances (degrees). 3. Convert the difference (degrees) to radians. 4. Plug the difference (radians) into s r
1 1 2 2 VII. Area of a Circle: A r, where θ is written in RADIANS! Sector a region bounded by 2 radii of the circle and their intercepted arc. Example: A car s rear windshield wiper rotates 125. The wiper mechanism wipes the windshield over a distance of 14 inches. Find the area covered by the wiper mechanism. Example: A sprinkler on a golf course fairway is set to spray water over a distance of 7 feet and rotates through an angle of 12. Find the area of the fairway watered by the sprinkler.
11 VIII. Linear Speed ( ) Measures how fast a particle moves along the circular arc of radius r. Movement is assumed to be a constant value/speed. Think length time miles hour cm sec arc _ length s r time t t, where θ is in RADIANS! Angular Speed ( ) How fast the angle changes when a particle moves along ( is swept out ) the circular arc of radius r at a constant speed. Think radians time or revolutions time (AND THEN CONVERT REVOLUTINS TO RADIANS USING 2 radians 1revolution ) central _ angle time t, where θ is in RADIANS! Comment:
12 Example: The second hand of a clock is 1.2 cm long. Find the linear speed of the tip of the second hand as it passes around the face of the clock. Example: My truck tires have a 2 radius and turn at 4 revolutions per second. How fast is my truck moving? (Write the final answer in miles per hour). Example: A Ferris wheel with a 1 foot diameter makes 1.5 revolutions per minute. A) Find the angular speed of the Ferris wheel in radians per minute. B) Find the linear speed of the Ferris wheel.
13 Section 7.2 and 7.3 and 7.5 Right Triangles Let θ be an angle in standard position with (x, y) a point on the terminal side of θ and r = 2 2 x y. For 9 / 2 : sin opp b y hypot c r csc hypot c r opp b y cos adj a x hypot c r sec hypot c r adj a x tan opp b y adj a x cot adj a x opp b y Example: 45-45-9 Triangles
14 Example: 3-6-9 Triangle Let s put the exact trigonometric values for the 45-45-9 triangle and the 3-6- 9 triangle in a table. θ in radians 6 θ in degrees 3 sin θ cos θ 4 45 3 6
15 Let s also put the exact trigonometric values for the 45-45-9 triangle, the 3-6- 9 triangle, and the quadrant angles on the unit circle. We have been thinking of the trigonometric functions as functions of an angle. Now let s think of the trigonometric functions as functions of a real number. Imaging wrapping the real number line around a unit circle (radius = 1). Wrapping counterclockwise produces a positive number. Wrapping clockwise produces a negative number. Since r = 1: The arc intercepted by the angle θ has length Note: This is section 7.5 in the text. Also, the text uses the notation (a, b) instead of (x, y). So just let x = a and y = b if you want to match with the text.
16 The Unit Circle
17 Trigonometric Functions of Quadrant Angles Find the exact values of each of the six trigonometric functions at a) b) 9 2 c) 18 d) 3 27 2
18 Reciprocal Identities Quotient Identities sin 1 csc csc 1 sin tan sin cos cos 1 sec sec 1 cos cot cos sin tan 1 cot cot 1 tan To find the five remaining trigonometric functions given one function: 1. Draw a picture and label the triangle. 2. Use the Pythagorean theorem to get the value of the remaining side. 3. Use the above identities to find the missing values and definitions.
19 Example: A resistor and inductor connected in a series network impede the flow of an alternating current. The impedance (z) is determined by the reactance (x) and 2 2 2 the resistance of the resistor (R): z x R ; all measured in ohms. is the phase angle. The resistance in a series is R = 588 ohms and the phase angle tan 5 12. satisfies a) Find the inductive reactance (x) and the impedance (z). b) Find the values of the trigonometric functions.
2 Pythagorean Identities 2 2 sin cos 1 2 2 tan 1 sec 2 2 cot 1 csc Proof: Example: Let θ be an acute angle such that sinθ =.6. Find the cosθ and tanθ using only identities.
21 Cofunctions of complementary angles are equal. Example: Use fundamental identities and complimentary angle theorem to find the exact value of: a) tan 2 cot 7 2 2 b) 1 cos 4 cos 5 c) cos7 sin2 cos2 sin7
22 Example: Let 2 functions in terms of x. and tanθ = x, where x. Express the trigonometric Example: Find an acute angle θ such that tanθ = cot (θ +45 ). Example: If tanθ = 4, find tanθ + tan( 2 ) exactly.
23 Examples: Use the unit circle to find: 2 a) cot tan 45 2csc 45 4 2 2 b) sin 3 cos 6 Example: Let f ( ) sin and g ( ) cos. Find a) b) g when 2 2 f when 2 3 3
24 III. A calculator will give approximate values (not exact) for the trigonometric functions. 1. Choose Mode. Choose degrees if there is a symbol. Choose radians if there is no symbol. 2. Sinθ, cosθ, and tanθ are on the calculator. 3. For cscθ, type in 1/sinθ For secθ, type in 1/cosθ For tanθ, type in 1/cotθ Example: A right triangle has one angle of 4. a) If the hypotenuse = 1cm, find the length of each leg. b) If one leg is 3 meters, what is the length of the hypotenuse? Why are there two answers?
25 IV. Angles of Elevation and Depression Angle of Elevation the acute angle from the horizontal up to the line of sight of the object. Angle of Depression the acute angle from the horizontal down to the line of sight of the object. Example: The Freedom Tower is 1776 feet tall. The angle of elevation from the base of an office building to the top of the tower is 34. The angle of elevation from the roof of the office building to the top of the tower is 2. A) How far is the office building from the Freedom Tower, measured to the nearest foot? B) How tall is the office building, to the nearest foot?
26 Comment: An angle of τ with the ground means to create this picture: Example: An 8 foot guy wire is attached to the top of a tower, making a 65 angle with the ground. How high is the tower? Section 7.4 Notice that the signs of the six trigonometric functions depend on the quadrant that you are in:
27 Example: Name the quadrant that you are in when: a) sinθ<, cosθ > b) cosθ>, tanθ > c) sinθ<, cotθ> d) cscθ>, cotθ< Trigonometric functions for any angle (not just acute angles) Let θ be an angle in standard position with (x, y) a 2 2 point on the terminal side of θ and r = x y. 1. Determine the quadrant that θ lies in. 2. Draw your triangle. 3. Use the Pythagorean theorem to find r. 4. Find the values of the trigonometric functions. 5. Attach the correct sign.
28 Example 1: Let (-3, 4) be a point on the terminal side of θ. Find the exact value of the six trigonometric functions. Example 2: Let tanθ = 5 4 and cosθ >. Find the exact value of the six trigonometric functions.
29 Reference Angle Let θ be an angle in standard position. It s reference angle is the acute angle formed by the terminal side of θ and the horizontal axis. Quadrant I Quadrant II Quadrant III Quadrant IV Example: Find the reference angle ' for each angle below: a) θ=3 b) θ=-135 c) θ= 2 3 d) θ= 19 6
3 Steps to find the trigonometric functions for any angle with reference angle : (not just angles in standard position) 1. Determine the quadrant that θ lies in. 2. Find the reference angle θ for θ 3. Find the values of the six trigonometric functions: sin θ cos θ tan θ csc θ sec θ cot θ 4. Assign the appropriate signs. sin θ=±sin θ cos θ=±cos θ tan θ=±tan θ Example: Find the exact values of the trigonometric function a) cos 4 3 b) tan 21 c) csc 11 4 d) sec 54
31 Example: An object is propelled upwards at angle θ, 45 9, to the horizontal with an initial velocity of v feet per second from the base of a plane that makes a 45 angle with the ground. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the function 2 v 2 R sin 2 cos 2 1. 32 1. Find the distance R that the object travels along the inclined plane if the initial velocity is 32 feet per second and 6 2. Graph R = R(θ) if the initial velocity is 32 ft/sec. 3. What value of θ makes R the largest?
32 The Rest of Section 7.5 FUNCTION SYMBOL DOMAIN RANGE f sin f cos f tan f csc f sec f cot
33 A function is periodic if there exists a positive real number θ such that f p f for all θ in the domain of f. The smallest number p for which f is periodic is called the period of f. What is the period of the six trigonometric functions? Periodic Properties: Let k be any Integer. sin 2 k sin tan k tan cos 2 k cos cot k cot csc 2 k csc sec 2 k sec Example: Find sin 39 using periodic properties.
34 A function f is even if f f : cos cos sec sec Example a) cos 27 b) sec 3 c) cos 17 4 A function f is odd if f f : sin sin csc csc tan tan cot cot Example a) sin 135 b) sin 3 2 c) tan 6 d) cot 7 2
35 Section 7.6 I. Sinusoidal / Sine Function Overall Graph: Close Up:
36 Cosine Function Overall Graph: Close Up:
37 Properties of the sine function: Period: 2 Domain: Set of Real numbers Range: [-1, 1] Symmetric about the origin (odd function) x-intercepts:, -2π, - π,, π, 2 π, 3 π, Maximum is y = 1. This occurs at x =, Minimum is y = -1. This occurs at x =, Properties of the cosine function: Period: 2 Domain: Set of Real numbers Range: [-1, 1] Symmetric about the y-axis (even function) x-intercepts:, 3 3,,, 2 2 2 2, 3 5 9,,, 2 2 2 2, 3 7,,,... 2 2 2 Maximum is y = 1. This occurs at x =, 2,,2,4,6, Minimum is y = -1. This occurs at x =,,,3,5, Comment: Notice that the graph of f(x) = sin(x) is g(x) = cos(x- are just shifted by 2. ). The graphs 2 sin(x) = cos(x- 2 )
38 III. Amplitude: For y = Asin(x) and y = Acos(x), the amplitude (A) represents the half-distance between the maximum and minimum values of the function. Amplitude = A Amplitude = max imum min imum 2 A > 1 => vertical stretch < A < 1 => vertical compression Example: Graph y = 2sin(x) over the interval [ -π, 4 π]
39 Example: Graph y = 2sin(-x) over the interval [ -2 π, 2 π] Example: Graph y = - ½cos(-x) over the interval [ -2 π, 2 π]
4 IV. The period of y = Asin( x) and y = Acos( x) is T = 2. < < 1 => horizontal stretch (period is greater than 2π) > 1 => horizontal shrink (period is less than 2π) Example: Graph y = sin( 2 x ) Example: Graph y = -3cos(-2x)
41 V. Vertical Translation For y = Asin( x) + B and y = Acos( x) + B, the graph is shifted: B units up for B > B units down for B < The graph now oscillates about the line y = B instead of about the line y = (the x-axis). Example: Graph y = 3cos(4x) + 2 Example: Write the equation of the graph below using a cosine function.
42 Example: Find the equation of the graph below. Example: Write the equation of a sine function with amplitude four and period one.
43 Example: Graph y = -3sin(-2x) + 1
44 Section 7.7 I. Cosecant Function Overall Graph: Close Up:
45 II. Secant Function Overall Graph: Close Up:
46 Steps to graph y = Acsc( x) + B Replace csc by sin Perform all transformations and graph y = Asin( x) + B Take the reciprocal of the y-coordinate to obtain y = Acsc( o Maximum on sine = relative minimum on csc o Minimum on sine = relative maximum on csc o X-intercepts = vertical asymptotes x) + B Steps to graph y = Asec( x) + B Replace sec by cos Perform all transformations and graph y = Acos( x) + B Take the reciprocal of the y-coordinate to obtain y = Asec( o Maximum on cos = relative minimum on sec o Minimum on cos = relative maximum on sec o X-intercepts = vertical asymptotes x) + B
47 Example: Graph each of the functions below. State the domain, the range, and the intervals that the functions are increasing and decreasing. a) y = 3csc( 3 2 x) b) y = csc(2x) + 2 c) y = 3sec(x) 2 d) y = 3sec( 1 4 x) e) y = -2sec(4x)
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49 III. Tangent Function Overall Graph: Close Up:
5 Graphing Y = Atan( x) + B: B is the vertical shift Amplitude (A): Think vertical stretch. There is not really a concept of amplitude for the tangent/cotangent functions. o Moves the points ( 4, -1) and ( 4, 1) to ( 4, - A ) and ( 4, A ). o Consider A = -1: y = -tax (x) = tan (-x) This flips the graph: Period = T = o Left asymptote can be found by solving x 2 o Right asymptote can be found by solving x 2 o Midpoint between the two asymptotes is the x intercept.
51 Example: Graph y = 3tan(-x). State the domain and range. Example: Graph y = 3tan( x ) 2.
52 IV. Cotangent Function Graphing y = Acot( x) + B: A is a vertical stretch. ( 4, A) and ( 3 4, - A ) Consider A = -1: y = cot(-x) = -cot(x) This flips the graph. Period: t = B is the vertical shift
53 Example: Graph y = 2cot(x) 1 Example: Graph y = 3cot( 2 x ) Comment: tan(x) = - cot(x+ 2 ). The graphs of these two equations are identical.
54 Section 7.8 Phase Shifts on the Cosine and Sine Curves y = Asin( x ) + B y = Acos( x ) + B Steps to Graph: 1) Write in factored form. y = Asin x + B y = Acos x + B Example: Write each equation in factored form. a) y = 3sin(3x-π) + 1 b) y = - 2cos(2x- 2 ) 2) Determine the amplitude A 2 3) Determine the period T = T 4) Divide the period into four equal intervals: 4 5) Apply the phase shift Move the graph right by for (x - ) Move the graph left by for (x + ) 6) Apply the vertical shift of B
55 Example: Graph a) y = 3sin(3x-π) + 1 b) y = - 2cos(2x- 2 )
56 Example: Write the equation of a sine function with Amplitude: 2 Period: Phase Shift: - 2
57 Example: The current I, in amperes, flowing through an alternating current circuit at time t (in seconds) is I( t) 12sin 3, t 3 What is the period? Amplitude? Phase shift?
58 Example: Throughout the day, the depth of water at the dock varies with the tides. The table shows the depth (in feet) at various times during the morning. Time (t) Depth (y) Midnight 3.4 2am 8.7 4am 11.3 6am 9.1 8am 3.8 1am.1 noon 1.2 a) Use a trigonometric function to model the data. b) Find the depth of the water at 9:am.