Halldorson Honors Pre Calculus Name 4.1: Angles and Their Measures

Transcription

1 .: Angles and Their Measures. Approximate each angle in terms of decimal degrees to the nearest ten thousandth. a. θ = 5 '5" b. θ = 5 8'. Approximate each angle in terms of degrees, minutes, and seconds a. θ =.7 b. θ =.55. Convert the following angles to degree measure (decimal degrees) a. 7 b Convert the following angles to radian measure a. 08 b Solve each of the following word problems. Show all work. a. A clock in Copenhagen, Denmark makes one complete revolution every 5,75 years. The clock is 0 ft in diameter. How far does a point on the outside of the clock move every decade? b. A building in Milwaukee has a clock with a minute hand 0 feet long. What is the linear speed of the tip of the minute hand in in./sec? c. The Tour de France is a bicycle race held in France. The average speed of the cyclists in one race was 5 mph. The diameter of a wheel is approximately in. What is the average angular speed in radians/hr? d. A clock s pendulum is m long. From one side to the other it moves through an angle of 0. radians. Through what distance does the tip of the pendulum move? P age

2 . Find the values of all six trigonometric functions of the angle theta. a. b. θ 5 θ. Assume that theta is an acute angle in a right triangle. Find the five other trigonometric values of the angle theta. a. tanθ = b. 5 cscθ = 7. Solve the following triangles. Standard position and lettering is used. a. A= 8 50', b=. b. 7 A= ', a = 50 P age

3 8. Solve each of the following. Show all work. a. The largest tree in Sequoia National Park in California is called General Sherman. To find its height, park rangers marked off a distance of 85 feet from its base. At that point they found the angle of elevation to the top of the tree to be 55 50'. How tall is General Sherman? b. A housing developer wishes to create a triangular lot at the corner of a street. The sides bordering on the two sides of the corner are 5 ft and 0 ft, respectively. Determine the angles formed by the vertices of the lot. c. An airplane takes off from a runway at an angle of elevation of 0'. Two minutes later the plane is at an altitude of 000 ft. How far is the airplane from the point of takeoff? How fast is the plane flying in miles per hour? d. Suppose a laser beam is directed toward the visible center of the moon and it misses its assigned target by 0 seconds. Approximate how far it is, in miles, from its assigned target. Assume that the distance from the earth to the moon is,000 miles and that, because of the extreme distance, the surface of the moon is flat. P age

4 Review.-. Key. Use transformations to describe how the graph of the function is related to a basic trigonometric graph. x a. y = + cos( x ) b. y = sin Reflected over the x-axis Vertical stretch of Translated units down Translated units up horizontally stretched by translated units to the right Translated units to the right. Graph y sin( x) = over the interval x. a. State the period b. State the range [,]. Graph y sin = x over the interval x fx ( ) = sin( x) Determine the equation of the sinusoid shown below. x y = 0.5cos

5 5. Graph the tangent, cotangent, secant, and cosecant functions over the interval[, ] - - fx () = tan( x) - - sx ( ) = cos( x) sx ( ) = tan( x) - - sx () = sin() x - -. Graph the functions f ( x) =.5csc ( x) and f ( x) = tan( x) sx ( ) = -.5 sin( x) sx ( ) = -tan( x) For each of the following equations, determine i-iii. x x A.) y = + sin B.) y = x+ cos 5 i.) Is it periodic? A) yes B) no ii.) Is it a sinusoid? A) yes B) no iii.) Find the equations of two linear functions y and y such that y y y A) y = y = B) y = x y = x

6 Solve for x in the given interval: 8. sec( x) = ; x x = 0 9. csc( x) = ; x x = 7 0. csc( x) = 8; x x =. 7cot( x) = 9 ; x x =. Give the formula of a function that oscillates between the functions y= x 7 and y= x. y = x.5 +.5cos x or y = x.5 +.5sin x CALCULATORS ALLOWED. A person is seated on a Ferris wheel of radius 0 feet. The Ferris wheel makes one rotation every 0 seconds. The center of the wheel is 5 feet above the ground. Assume uniform speed from the beginning to the end of the ride. A passenger is half way to his maximum height when the ride begins. a. Express the person s height above the ground as a function of time. x y = 0sin b. Graph minutes of the ride. 0 height (feet) 0 fx () = 0 sin( x 5) time (minutes)

7 . Without graphing the following functions, determine which will be a sinusoid. If one is, find values for a, b, and h (to the nearest hundredth) so that: gx ( ) = asin[ bx ( h)] A.) f ( x) = sin( x) cos( x) B.) f ( x) sin( = x) +.5cos( x) A) Yes, gx ( ).0sin[( x 0.9)] B) Yes, gx ( ) =.80sin ( x+.0). Give the equation for the function with the following characteristics: x a. cot x : period of, vertical shift up f( x ) = cot + b. sec x: period of, phase shift to the right, vertical stretch of f( x) = sec ( x ). Determine the value(s) of k such that sec( kx ) has vertical asymptotes at every odd integer. 5. Solve for x is the given interval, support your answer (with a graph or algebraically) 5 7 A.) sec x= ; x 8 0 x =, B.) tan x= ; x k = C.) x csc x=.8; x x.87

8 Non Calculator. Sketch the graph of each function. a. f ( x) = x+ cosx b. f ( x) = xsin x 0 5 gx ( ) = x+ fx ( ) = x+cos( x) rx ( ) = -x sx ( ) = x sin x ( ) -0 hx ( ) = x c. x f ( x) = cosx d. f ( x) = cscx qx ( ) = x - vx ( ) = -x cos( x) tx ( ) = -x ux ( ) = - -x wx ( ) = sin x ( ) -. Verify that the function ( ) f ( x) = cos x = cosx is periodic. [ ] ( ) f ( x+ ) = cos( x+ ) = cos x = f( x). Decide if each of the following is a sinusoid. Then, find the period. a. f ( x) = sin5x cosx no, b. f ( x) = 7cosx 8sinx yes, P age

9 Calculator Part. Sketch a graph of each of the following functions. Decide if they are periodic, and if so, find the period. a. f ( x) = sinx+ cosx b. sin x f ( x) = e period = period = f x ( ) = sin x ( )+ cos x ( ) g ( x) = e si n( x) Approximate the following sum as a single sinusoid (round to the nearest hundredth). f ( x) = cosx sinx ( ) ( ) f( x) =.cos x+ 0. or f( x) =.cos x Solve the following problems. a. The profits of a company fluctuate during a month period according to the formula Pt ( ) = $50,000 sint+ cost (t is measured in months). ( ) i. Sketch a graph of t over the interval [0,] $ in thousands ii. 80 Explain the meaning of the negative values of the function. Intervals where the company lost money. P age

10 b. 0.5t The motion of a spring is given by D() t = 7e cos( t), where D(t)is measured in inches and time t is in seconds. a. Sketch a graph of D(t) on the interval [0,]. gx ( ) = 7 e -0.5 x fx ( ) = 7 e -0.5 x cos x ( ) 5 - hx ( ) = -7 e -0.5 x - b. What is the maximum value of the function? Y=7, when x=0 c. What is the minimum value of the function? Y= 5.9, when x=.87 P age

11 c. A spotlight 80 feet from a prison wall rotates in a circle once every seconds. At t=0 seconds, the spotlight is pointing directly at point Q. The spotlight is pointing at a distance d from Q after t seconds. a. Calculate the frequency of the rotation. revolution seconds b. Write a function for the d as a function of theta. d = 80tanθ c. Use the fact that θ = f t where f is the frequency (can you explain why?) in order to write a function for d as a function of t. t d = 80 tan spotlight θ B 80 ft d d. Sketch a graph of the function below x fx ( ) = 80 tan( ) e. Explain the meaning of the undefined values of d. The light is pointing along a perpendicular line to the wall f. Explain the meaning of the negative values of d. The light is pointing away from the wall. P age

12 Honors Pre Calculus Worksheet.7.8 Name A. Sketch a graph of each of the following functions, and write their domain and range:. f ( x) = tan ( x). f ( x) = cos ( x) D - :(, ) R :,. x f( x) = sin + 8. f ( x) = cos x.5 [ ] [ ] D :, R :0, [ ] [ + + ] D :, R :, B. Calculate the exact value for each of the following D :, R :0, [ ]. sin =. cos =. tan =. tan tan = 5. tan cos( ) =. ( ) cos tan = C. Find an expression for each of the following:. ( x) sin cos = x. 9x + ( x) csc tan = x

13 Honors Pre Calculus Worksheet.7.8 Name D. Solve each of the following problems.. An airplane takes off from a runway at an angle of elevation of 0'. Two minutes later the plane is at an altitude of 000 ft. How far is the airplane from the point of takeoff? 9 ft. A weight is attached to a spring and set in motion by compressing the spring from its rest position and releasing it. The maximum displacement of the spring is 8 cm, and it takes seconds to complete one cycle. a. Write an equation for the height of the weight as a function of time. ht () = 8cos t b. What is distance below the starting point of the spring after 5 seconds? 8 8cos 5 = 8 c. How far has the weight traveled during the first 5 seconds? 0 cm d. After how many seconds will the spring be 9 inches below the starting point?.08 seconds. A person s blood pressure changes in a rhythmic fashion, with a periodicity set by the besting of the heart. Suppose a person s blood pressure at time t minutes is given by the formula Pt ( ) = 0+ cos(0 t) a. Sketch the graph of P(t) below (show at least two full cycles of the periodic motion, as well as an accurate scale) b. Determine the rate at which the person s heart is beating (beats per minute) 80 c. What are the maximum and minimum blood pressure readings? Max = 5, min = 77 d. After how many minutes will the blood pressure be about 0? 0.00 sec