Section 8. Radians and Arc Length Definition. An angle of radian is defined to be the angle, in the counterclockwise direction, at the center of a unit circle which spans an arc of length. Conversion Factors: Degrees Radians Radians Degrees Example. Convert each of the following angles from radians to degrees or from degrees to radians. An angle measure is assumed to be in radians if the degree symbol is not indicated after it. (a) 0 (b) π (c).4 Example. On the unit circle to the right, label the indicated common angles with their degree and radian measures. Theorem. The arc length, s, spanned in a circle of radius r by an angle of θ radians, 0 θ π, is given by Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0
Examples and Exercises. In the pictures below, you are given the radius of a circle and the length of a circular arc cut off by an angle θ. Find the degree and radian measure of θ. 8 θ θ 4 4. In the pictures below, find the length of the arc cut off by each angle. π/ ο 80. A satellite orbiting the earth in a circular path stays at a constant altitude of 00 kilometers throughout its orbit. Given that the radius of the earth is 670 kilometers, find the distance that the satellite travels in completing 70% of one complete orbit. 4. An ant starts at the point (0,) on a circle of radius (centered at the origin) and walks units counterclockwise along the arc of the circle. Find the x and the y coordinates of where the ant ends up. Developed by Jerry Morris
Section 8. Sinusoidal Functions and Their Graphs y sinθ 70 90 90 70 450 60 Π Π Π Π Π Π Π 5 Π Π 7 Π Π 4Π Θ y cosθ 60 80 80 60 540 70 Π Π Π Π Π Π Π 5 Π Π 7 Π Π 4Π Θ Directions. Make sure that your graphing calculator is set in radian mode. Function Effect on y = sin x y = sinx y = sin x + y = sin(x + ) y = sin(x) B y = sin(bx) Period y = sin x y = sin(x) 4 y = sin(4x) / y = sin(x/) B y = sin(bx) Summary: For the sinusoidal functions y = A sin(b(x h)) + k and y = A cos(b(x h)) + k:. Amplitude =. Period =. Horizontal Shift = 4. Midline: Definition. A function is called sinusoidal if it is a transformation of a sine or a cosine function. Helpful Hints in Finding Formulas for Sinusoidal Functions. If selected starting point occurs at the midline of the graph, use the sine function.. If selected starting point occurs at the maximum or minimum value of the graph, use the cosine function.. Changing the sign of the constant A reflects the graph of a sinusoidal function about its midline. Example. Let y = sin(x π) +. Find the amplitude, period, midline, and horizontal shift of this function. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0
Example. Find a formula for the sinusoidal function to the right. 0 4 0 Example. Find a formula for the sinusoidal function to the right. 6 9 5 4 Developed by Jerry Morris
Examples and Exercises. Find a possible formula for each of the following sinusoidal functions. 6 Π Π Π Π 4Π 4 4 6 8 0.8 4 0.8 Π 7 0 Π Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0 5
. For each of the following, find the amplitude, the period, the horizontal shift, and the midline. (a) y = cos(πx + π ) (b) y = sin(x 7π). A population of animals oscillates annually from a low of 00 on January st to a high of 00 on July st, and back to a low of 00 on the following January. Assume that the population is well-approximated by a sine or a cosine function. (a) Find a formula for the population, P, as a function of time, t. Let t represent the number of months after January st. (Hint. First, make a rough sketch of the population, and use the sketch to find the amplitude, period, and midline.) (b) Estimate the animal population on May 5th. (c) On what dates will the animal population be halfway between the maximum and the minimum populations? 6 Developed by Jerry Morris
Section 8. Trig Functions: Relationships and Graphs Definitions. We define the secant, cosecant, and cotangent functions as follows: secθ = cosθ csc θ = sin θ cotθ = cosθ sinθ Preliminary Exercise. Shown below are graphs of y = secθ, y = cscθ, and y = cotθ (not necessarily in that order). Without using a graphing calculator, match each graph to the correct formula. (Suggestion: Use your knowledge of the behavior of the sine and cosine functions to make your matches; in particular, pay attention to where these functions are positive, negative, or zero.) (a) (b) (c) Π Π Π Π Π Π Π Π Π Π Π Π Example. Given to the right are graphs of the sine and cosine functions. First, label which is which and then complete the following: (a) Describe how the cosine function can be translated to obtain the sine function, and vice versa. Then, use your observations to complete identities () and () below. Π Π Π Π (b) Is the cosine function even, odd, or neither? How about the sine function? Use your observations to complete identities () and (4) below. Identities. ( ) () cos ( ) () sin = sin θ = cosθ () cos( θ) = (4) sin( θ) = Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0 7
Example. Illustrate Identity () from page 7 by filling in the ( table to the right. What do you notice? θ cosθ sin θ + π ) 0 More Identities π/6 π/4 π/ π/ Some observations: π θ θ (x, y) Identities. ( π ) (5) sin θ + cos θ = (6) cos θ ( π ) = sin θ (7) sin θ = cosθ Example. Given that sinθ = secθ. and that θ is in the second quadrant, find the exact values of cosθ, tanθ, and 8 Developed by Jerry Morris
Examples and Exercises. Suppose that sinθ = 4 and that π θ π. Find the exact values of cosθ and secθ.. Suppose that cscθ = x and that θ lies in the nd quadrant. Find expressions for cosθ and tanθ in terms of x. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0 9
Section 8.4 Trig Equations and Inverse Functions Example. Consider the graph of y = cosθ given to the right. (a) Graphically estimate the solutions to cos θ = 0., where π θ π. (Note: Angles, θ, are on the horizontal axis) 0 6 4 0 4 6 0 6 4 0 4 6 (b) Use your calculator to find cos ( 0.), and compare with your answers to (a). (c) Use your calculator to find cos (0.), and use this number and symmetry to obtain more accurate solutions to cosθ = 0.. Example. For each equation, find all solutions between 0 and π, giving exact solutions where possible. (a) sin x = 7 0 (b) 6 cosx = 0 Developed by Jerry Morris
Example. Find all solutions between 0 and π to tanx = exactly. Example 4. The height of a rider on a ferris wheel is given by h(t) = 0 cos(πt) meters, where t gives time, in minutes, after the person boards the ferris wheel. (a) Find the amplitude, midline, and period of the function h, and then sketch a graph h corresponding to the first two minutes of the ride. (b) During the first two minutes of the ride, find the times when the rider has a height of 0 meters. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0
Examples and Exercises. Find all solutions between 0 and π to each of the following equations, giving exact solutions where possible. (a) sin x = (b) tanx 5 = (c) cosx =. An animal population is modeled by the function ( π ) P = 00 + 00 sin 6 t 500 P (see graph to the right), where t represents the time, in days, after the beginning of January st. Find the times at which the animal population is 50. 6 t Developed by Jerry Morris
. In this problem, you will investigate the general definition of the inverse cosine and inverse sine functions. (a) Use your calculator to fill in the following blanks, and then plot the corresponding points on the provided graph of y = cosθ. Between what two numbers do all the θ values appear to lie? 4 6 Θ cos ( ) = cos ( 0.8) = cos ( 0.4) = cos (0) = cos (0.4) = cos (0.8) = cos () = (b) Use your calculator to fill in the following blanks, and then plot the corresponding points on the provided graph of y = sin θ. Between what two numbers do all the θ values appear to lie? 4 6 Θ sin ( ) = sin ( 0.8) = sin ( 0.4) = sin (0) = sin (0.4) = sin (0.8) = sin () = Definitions. () We define the inverse cosine or arccosine function as follows: cos y is the angle between 0 and π whose cosine equals y. () We define the inverse sine or arcsine function as follows: sin y is the angle between π and π whose sine equals y. () We define the inverse tangent or arctangent function as follows: tan y is the angle between π and π whose tangent equals y. Activities to accompany Functions Modeling Change, Connally et al, Wiley, 0