Unit 8 Trigonometry Math III Mrs. Valentine
8A.1 Angles and Periodic Data * Identifying Cycles and Periods * A periodic function is a function that repeats a pattern of y- values (outputs) at regular intervals. * One complete pattern = cycle * Horizontal length of one pattern = period * Example: Analyze the periodic function below. Identify the cycle in two different ways. What is the period of the function? Each cycle is highlighted in red. Note that you can start in multiple places to indicate a cycle, but the period is the same. The period of this function is 4.
8A.1 Angles and Periodic Data * Identifying Periodic Functions * Analyze each graph to determine if the function is periodic. If it is, find the period. No Yes, 4 Yes, 8 No
8A.1 Angles and Periodic Data * Finding Amplitude and Midline of a Periodic Function * The amplitude of a periodic function measures the amount of variation in the function values. * Midline: horizontal line midway between the maximum and minimum of the function * Amplitude: half the distance between the minimum and maximum. Amplitude = ½ (maximum value minimum value) * Example: What is the amplitude of the periodic function? What is the equation of the midline? Amplitude = ½ (4 ( 2)) = ½ (6) = 3 Midline = ½ (4 + ( 2)) y = ½ (2) y = 1
8A.1 Angles and Periodic Data * Using a Periodic Function to Solve a Problem * Some data can be modeled using periodic functions, such as heartbeats, the cycles of a Ferris wheel, etc. * Example: Sound is produced by periodic changes in air pressure called sound wave. The yellow graph in the digital wave display at the right shows the graph of a pure tone from a tuning fork. What are the period and the amplitude of the sound wave? One cycle: from 0.004 to 0.008 Period = 0.008 0.004 = 0.004 Amplitude = ½ (2.5 1.5) = ½ (1) = ½ The period of the sound wave is 0.004s. The amplitude is ½.
8A.1 Angles and Periodic Data * Measuring Angles in Standard Position * An angle in the coordinate plane is in standard position when the vertex is at the origin and one ray is on the positive x- axis. * Initial side is on the x-axis. * Terminal side is the other ray of the angle. * The measure of the angle in standard position is the input for two important functions: cosine and sine. * The measure of the angle is positive when the rotation is counterclockwise and negative in the clockwise direction.
8A.1 Angles and Periodic Data * Measuring Angles in Standard Position * Examples: What are the measures of each angle? Counterclockwise 90 Clockwise - 90 + (- 45 ) = - 135 Counterclockwise 180 + 45 = 225 Clockwise - 270 + (- 45 ) = - 315
8A.1 Angles and Periodic Data * Sketching Angles in Standard Position * What is a sketch of each angle in standard position? * 36 * 315 * 150
8A.1 Angles and Periodic Data * Identifying Coterminal Angles * Coterminal angles are two angles in standard position with the same terminal side. * Which of the following angles is not coterminal with the other three: 300, 60, 60, 420? 300, 60, and 420 all have a terminal side in quadrant IV (the same terminal side) while 60 has a terminal side in quadrant I. So 60 is not coterminal with the others.
8A.2 The Unit Circle and Radians * Finding Cosines and Sines of Angles * In a 360 angle, a point 1 unit from the origin on the terminal ray makes one full rotation around the origin à UNIT CIRCLE * Any right triangle formed by the radius of the unit circle has a hypotenuse of 1. * Suppose an angle in standard position has measure θ. * Cosine of θ (cosθ) is the x-coordinate of the point at which the terminal side of the angle intersects the unit circle. * Sine of θ (sinθ) is the y-coordinate.
8A.2 The Unit Circle and Radians * Example: What are cos θ and sin θ for θ = 90, θ = 180, and θ = 270? Cos (90 ) = 0 Sin (90 ) = 1 Cos ( 180 ) = 1 Sin ( 180 ) = 0 Cos (270 ) = 0 Sin (270 ) = 1
8A.2 The Unit Circle and Radians * Finding Exact Values of Cosine and Sine * You can find exact values of sine and cosine for angles that are multiples of 30 and 45. * Example: What are the cosine and sine of θ = 60? The cosine of 60 is the length of the shorter leg of the right triangle formed using the radius at 60. The sine of 60 is the length of the longer leg. Recall that in a 30-60 -90 triangle: cos (60) = ½ (1) = ½ sin(60) = (3) * ½ = (3) / 2
8A.2 The Unit Circle and Radians * Example: What are the cosine and sine of θ = 225? Recall that in a 45-45 -90 triangle: The legs are equal to each other Since this is in quadrant III, both x and y should be negative.
8A.2 The Unit Circle and Radians * Using Dimensional Analysis * Central angle angle with vertex at the center of a circle. * Intercepted arc portion of circle between the endpoints of a central angle. * Radian measure of a central angle that intercept an arc with length equal to the radius of the circle. * Use to convert between degrees and radians. * Examples: Convert to degrees Convert 27 to radians = 135
8A.2 The Unit Circle and Radians * Finding Cosine and Sine of a Radian Measure * What are the exact values of and? This creates a 45-45 -90 triangle using the radius as the hypotenuse. Therefore,
8A.2 The Unit Circle and Radians * Finding the Length of an Arc * The length (s) of an intercepted arc is s = rθ where r is the radius and θ is the angle in radians. * Example: Use the circle below. What is length s to the nearest tenth? What is the length of b? r = 3in θ = 5π / 6 s = rθ s = 3( 5π / 6 ) s = 5π / 2 = 7.9 in. r = 3in θ = 2π / 3 b = rθ b = 3( 2π / 3 ) b = 2π = 6.3 in.
8A.2 The Unit Circle and Radians * Using Radian Measure to Solve a Problem * A weather satellite in a circular orbit around Earth completes one orbit every 2 h. How far does the satellite travel in 1 h? Angle for 1h of travel: Find the length of the arc:
8A.2 The Unit Circle and Radians * The Unit Circle in Radians It is highly important to know your unit circle. This one shows measure of angles in both degrees and radians, as well as the cosine and sine of each angle. You will be expected to commit this unit circle to memory (see handout).
8A.3 Sine & Cosine Functions * Estimating Sine Values Graphically * The sine function, y = sinθ, matches the measure of angle θ of an angle in standard position with the y-coordinate of a point on the unit circle. * It is much easier to graph in radians than in degrees for sine functions.
8A.3 Sine & Cosine Functions * What is a reasonable estimate for each value from the graph? Check your estimate with a calculator. sin 2 The sine function reaches is maximum at π/2. sin 2 is slightly past that, so it is about 0.9 Check: sin 2 = 0.9092974268 sin π The sine function crosses the x-axis at π, so sin π = 0 Check: sin π = 0
8A.3 Sine & Cosine Functions * Finding the Period of a Sine Curve * The graph of a sine function is called a sine curve. * Examples: For the graph of y = sin 4x, how many cycles occur in the window described? Window: There are 4 cycles What is the period of y = sin 4x? Divide the interval by the number of cycles. The period of y = sin 4x is π/2 2π / 4 = π/2
8A.3 Sine & Cosine Functions * Finding the Amplitude of a Sine Curve * You can also vary the amplitude of a sine wave. * Example: The graphs show y = a sinx. Each x-axis shows values from 0 to 2π. What is the amplitude of each graph? a = ½ (3 (-3)) = ½ 6 = 3 a = ½ (0.6 (-0.6)) = ½ 1.2 = 0.6
8A.3 Sine & Cosine Functions * Sketching a Graph * Properties of sine functions: * What is the graph of one cycle of a sine curve with amplitude 2, period 4π, midline y = 0, and a >0? Using the form y = a sin b θ, what is the equation for the sine curve? a = 2 4π = 2π/b b = ½ y = 2 sin ½θ
8A.3 Sine & Cosine Functions * Graphing From a Function Rule * What is the graph of one cycle of y = ½ sin 2θ? a = ½ = ½ b = 2, so it cycles 2 times from 0 to 2π Period: 2π/ b = 2π / 2 = π * What is the graph of one cycle of y = 3 sin π/2 θ? a = 3 = 3 b = π/2, so it cycles π/2 times from 0 to 2π Period: 2π/ (π/2) = 4
8A.3 Sine & Cosine Functions * Using the Sine Function to Model Light Waves * The graphs provided model waves of red, blue, and yellow light. What equation best models each color light? Blue: a = 1 Period = 480 = 2π/b b = 2π / 480 = π / 240 y = sin π/240θ Red: a = 1 Period = 640 = 2π/b b = 2π / 640 = π / 320 y = sin π/320θ Yellow: a = 1 Period = 570 = 2π/b b = 2π / 570 = π / 285 y = sin π/285θ
8A.3 Sine & Cosine Functions * Interpreting a Graph * The cosine function, y = cosθ, matches θ with the x-coordinate of the point on the unit circle where the terminal side of angle θ intersects the unit circle. * The symmetry of the set of points (X, y) = (cos θ, sin θ) on the unit circle guarantees that the graphs of sine and cosine are congruent translations of each other.
8A.3 Sine & Cosine Functions * Example: What are the domain, period, range, and amplitude of the cosine function? The domain of the function is all real numbers The function goes from its maximum value of 1 to its minimum value of -1 and back again in an interval from 0 to 2π. The period is 2π. The midline is y = 0. The range of the function is -1 y 1 Amplitude = ½ (max min) = ½ (1 (-1)) = 1 * Where in the cycle do the maximum and minimum occur? The zeros? The maximum value occurs at 0 and 2π. The minimum value occurs at π. The zeros occur at π/2 and 3π/2.
8A.3 Sine & Cosine Functions * Sketching the Graph of a Cosine Function * Properties of Cosine Functions: * To graph a cosine function, locate five points equally spaced through one cycle. For a > 0, this five-point pattern is max-zero-min-zero-max. * Example: Sketch one cycle of y = 1.5 cos 2θ. a = 1.5 = 1.5 b = 2, so it cycles 2 times from 0 to 2π Period: 2π/ b = 2π / 2 = π
8A.3 Sine & Cosine Functions * Modeling with a Cosine Function * The water level varies from low tide to high tide as shown. What is a cosine function that models the water level in inches above and below the average water level? Express the model as a function of time in hours since 10:30AM. Amplitude is ½ (60) = 30. Since the tide is at -30 inches at time 0, the curve follows min-zero-max-zero-min pattern, so a = 30. The cycle is half-way complete after 6 h and 10 min, so the full period is 12 hours and 20 minutes, or 12⅓ hours. 12⅓ = 2π/b b = 6π/37 f(t) = 30 cos ((6π/37)t)
8A.3 Sine & Cosine Functions * Solving a Cosine Equation * You can solve an equation by graphing to find an exact location along a sine or cosine curve. * Example: Suppose you want to find the time t in hours when the water level from the last problem is exactly 10 in. above the average level represented by f(t) = 0. What are all the solutions to the equation 30 cos ((6π/37)t) = 10 in the interval from 0 to 25? Y 1 = 30 cos ((6π/37)t) Y 2 = 10 Find the points of intersection t 3.75, 8.58, 16.08, and 20.92 The water level is 10in above average at about 3.75h, 8.58h, 16.08h, and 20.92h after 10:30AM
8A.4 The Tangent Function * Finding Tangents Geometrically * The tangent function is closely associated with sine and cosine but is different from them in three dramatic ways: 1. The tangent function has infinitely many points of discontinuity with vertical asymptotes at each one 2. Range = all real numbers; domain = all real numbers except odd multiples of π/2. 3. Its period is π, half that of sine and cosine * For any angle in the unit circle where point (x,y) is the point of intersection between the terminal side and the unit circle, tangent of θ, tanθ, is the ratio y / x.
8A.4 The Tangent Function * Example: What is the value of each expression? tan π An angle of π radians in standard position has a terminal side that intersects the circle at ( 1, 0). tan π = 0 / 1 = 0 tan (-5π/6) An angle of -5π/6 radians in standard position has a terminal side that intersects the circle at ( (3)/2, -1/2). tan -5π/6 = (-1/2) / (- (3)/2) = 1/ (3) = (3)/3
8A.4 The Tangent Function * Graphing a Tangent Function * Another way to find the tangent of an angle is to draw the tangent line to the unit circle, x = 1, and see where the terminal side of that angle will intersect with it when extended. The y-coordinate of that point is the tangent of that angle. * The graph at the right shows one cycle of the tangent function, y = tan θ, for π/2 < θ < π/2.
8A.4 The Tangent Function * Properties of tangent: * You can use asymptotes and three points to sketch one cycle of a tangent curve. * Use the pattern asymptote-( a)-zero-(a)-asymptote * The period can help to determine the positions of the asymptotes.
8A.4 The Tangent Function * Example: Sketch two cycles of the graph of tan πθ. Period: π/b = π/π = 1 Cycle: -π/2b = -π/2π = -½ π/2b = π/2π = ½ Asymptotes are at θ = -½, ½, and 3 / 2 Divide period into fourths and locate 3 points between the asymptotes for each cycle. Points: ( -¼, -1), (0,0), (¼, 1) ( ¾, -1), (1,0), ( 5 / 4, 1)
8A.4 The Tangent Function * Using the Tangent Function to Solve Problems * An architect is designing the front façade of a building to include a triangle, similar to the one shown. The function y = 100 tan θ models the height of the triangle, where θ is the angle indicated. Graph the function using the degree mode. What is the height of the triangle if θ = 16? If θ = 22? Graph the function. Use the table feature to find the heights. If θ = 16, height is about 28.7ft. If θ = 22, height is about 40.4ft.
8A.5 Translating Sine and Cosine Fns * Identifying Phase Shifts * For any function f, you can graph f(x h) by translating the graph of f by h units horizontally. * Each horizontal translation of certain periodic functions is a phase shift. * f(x) + k will translate the graph of f by k units vertically. * Each vertical translation of certain periodic functions is a midline shift.
8A.5 Translating Sine and Cosine Fns * Examples: What is the value of h in each translation? Describe each phase shift (use a phrase such as 3 units to the left). g(t) = f(t 2) y = cos(x + 4) y = sin (x + π) h = 2 The phase shift is 2 units to the right h = 4 The phase shift is 4 units to the left h = π The phase shift is π units to the left
8A.5 Translating Sine and Cosine Fns * Graphing Translations * You can analyze a translation to determine how it relates to the parent function. * Example: Use the graph of the parent function y = sin x. What is the graph of each translation in the interval 0 x 2π? y = sin x + 3 k = 3 The midline shift is 3 units up. The new midline is y = 3. y = sin (x π / 2 ) h = π / 2 The phase shift is π / 2 units to the right
8A.5 Translating Sine and Cosine Fns * Graphing a Combined Translation * You can translate both vertically and horizontally to produce combined translations. * Example: Use the graph of the parent function y = sin x. What is the graph of the translation y = sin (x + π) 2 in the interval 0 x 2π? h = π The phase shift is π units to the left. k = 2 The midline shift is 2 units down. The new midline is y = 2.
8A.5 Translating Sine and Cosine Fns * Graphing a Translation of y = sin 2x * Families of Sine and Cosine Functions: * Example: What is the graph of y = sin 2(x π / 3 ) 3 / 2 in the interval from 0 to 2π? Sketch the original graph of y = sin 2x Translate each of the points π / 3 to the right and 3 / 2 down.
8A.5 Translating Sine and Cosine Fns * Writing Translations * What is an equation that models each translation? y = sin x, π units down k = π New Equation: y = sin x π y = - cos x, 2 units to the left h = 2 New Equation: y = - cos (x + 2) y = sin x, π / 2 units right and 3 units down h = π / 2 k = 3 New Equation: y = sin (x π / 2 ) 3
8A.5 Translating Sine and Cosine Fns * Writing a Trigonometric Function to Model a Situation * The table give the average temperature in your town x days after the start of the calendar year ( 0 x 365). Make a scatter plot of the data. What cosine function models the average daily temperature as a function of x? Amplitude = ½ (max min) = ½ (77 33) = 22 Period = 365 = 2π / b, so b = 2π / 365 y = 22 cos 2π / 365 (x 198) + 55 Phase shift: h = 198 0 = 198 Vertical shift: k = 77 22 = 55