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Practice Questions 1. If s denotes the length of the arc of a circle of radius r subtended by a central angle ΞΈ, find the missing quantity: a. r = 6.22 centimeters, ΞΈ = 1.8 radians, s = 11.2cm b. r= 23 feet, s=14 feet, ΞΈ = 0.609 radians 2. The minute hand of a clock is 7 inches long. How far does the tip of the minute hand move in 5 minutes? If necessary, round the answer to two decimal places. 3.67 in 3. An irrigation sprinkler in a field of lettuce sprays water over a distance of 30 feet as it rotates through an angle of 120°. What area of the field receives water? If necessary, round the answer to two decimal places. 942.48 ft2 4. A pick-up truck is fitted with new tires which have a diameter of 40 inches. How fast will the pick-up truck be moving when the wheels are rotating at 445 revolutions per minute? Express the answer in miles per hour rounded to the nearest whole number. 53 mph For 5 and 6 : two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. 5. Find csc B when a = 7 and b = 2. βππ 6. Find cot A when a = 6 and c = 7. βππ π β7 4 π 3 7. If sinΞΈ= , cosΞΈ= then find all the other trig functions. (Use 4 fundamental identities.) βπ π πβπ πβπ π π π π tanΞΈ= , cotΞΈ= 8. Find the value of β , cscΞΈ= sin 20° cos 70° , secΞΈ= . Do not use a calculator. -1 9. A barge is located 200 feet away from the coastline 1200 feet down the coast from a power source at point A. To supply the barge with electricity, a power line will be run from point A to a point B on the coast and then from point B to the barge. If power lines on land coast $3 per foot and power lines under water cost $5 per foot, calculate the total cost of running a power line from point A to the barge when point B is 800 feet down the coast from point A in the direction of the barge. Recalculate the total cost when the distance from A to B is increased in 50-foot increments, until you locate a possible minimum cost. Interpret your solution. $4636.07; The minimum cost is $4400 when the point is chosen 1050 feet to the right of A. Part I: calculate the total cost of running a power line from point A to the barge Let distance (A to B) = L and distance (B to barge) = W Cost function = 3*L + 5*W = 2400 + 5*W (since L = 800) We can express W =β2002 +4002 =447.2136 Cost =3*800 + 5*447.2136 =4636.07 Part II: Recalculate the total cost when the distance from A to B is increased in 50-foot increments, until you locate a possible minimum cost C ΞΈ Letβs rewrite our Cost function = 3*(800 + P) + 5Q = 2400 + 3P + 5Q Where P is the amt you increase the power lines on land (in addition to 800 ft) And Q is the length of the power line in water. To achieve the minimum value you need to increase ΞΈ such that the cost function is minimized. The distance A to B is 800 and B to C is 400. Using right angle trig I can express Q and P as 200 200 Q= , P = 400π πππ π‘πππ Cost Function = 2400 +3*(400 = 3600 - 600 π‘πππ + 200 π‘πππ 1000 π πππ )+ 5* 500 π πππ Plotting in excel we our cost function looks as below: Radians Cost function 4650 4600 Cost function 0.46 4636.07 0.50 4587.54 0.60 4494.01 0.70 4439.93 4550 0.80 4411.28 4500 0.90 4400.48 4450 1.00 4403.14 4400 1.10 4416.69 1.20 4439.65 1.30 4471.25 1.40 4511.28 4350 0 0.5 1 1.5 2 1.50 4559.96 1.60 4617.95 1.57 4600.00 10. A building 260 feet tall casts a 70 foot long shadow. If a person looks down from the top of the building, what is the measure of the angle between the end of the shadow and the vertical side of the building (to the nearest degree)? (Assume the personΚΉs eyes are level with the top of the building.) 15° 5 β11 ) 6 6 11. The point ( , on the unit circle corresponds to a real number t is given. Find all six trig functions. βππ , ππ¨π¬π π sinΞΈ= π βππ , π = , tanΞΈ= π cotΞΈ= π , cscΞΈ= βππ 12. Use a coterminal angle to find a. cos 765° = b. csc 41π 4 βπ , π = βπ 13. Name the quadrant in which the angle ΞΈ lies: a. tan ΞΈ > 0, sin ΞΈ < 0 - Quadrant III b. csc ΞΈ > 0, sec ΞΈ > 0 - Quadrant I π π , secΞΈ= βππ π Find the reference angle of the given angle and all six trig functions for each of the following: a. 100 ° 80° (use calculator to find trig funcs) b. -391 ° 31° (use calculator to find trig funcs) 41π π π π π π c. (using unit circle, sin =βπ, cos = , tan =π, 14. 4 π π π π π βπ π π π π βπ π csc = , sec =βπ , tan =1) π A study of ice cream consumption over 30 four-week periods in the early 1950s gives rise to the equation C = 0.3520 + 0.0786 sin(0.4806ΞΈ - 0.1691) where ΞΈ is the number (from 1 to 30) of the four-week period and C is the ice cream consumption in pints per capita. Determine the consumption for the tenth four-week period. Round answer to the nearest 0.001 pint. 0.274 pint 15. 16. If f(ΞΈ) = tan ΞΈ and f(a) = -3, find the exact value of f(-a). 3 17. Find the exact value of the expression. Do not use a calculator. a. sin (2Ο) + cos (β 5π b. csc ( ) sec (β 2 18. 5π 2 7π 3 )=0 ) =2 Use transformations to graph the following functions: a. y = 3 sin x + 5 b. y = cos (x - Ο) π c. y = -tan(π₯ + ) 1 2 π d. y = cot(π₯ + ) 2 4 π e. y = csc(π₯ β ) 4 19. What is the y-intercept of : a. y = tan x? 0 b. y = csc x? not defined 20. The data below represent the average monthly cost of natural gas in an Oregon home. 26) The data below represent the average monthly cost of natural gas in an Oregon home. M onth Cost A ug 21.20 Sep 28.24 M onth Feb Cost 111.30 M ar 106.26 Oct 44.73 A pr 89.77 N ov 67.25 M ay 67.25 Dec 89.77 Jun 43.73 Jan 106.26 Jul 28.24 Above is the graph of 45.05 sin x superimposed over a scatter diagram of the data. Find the sinusoidal function of the form y = A sin (Οx - Ο) + B which best fits the data. A bove is the graph of 45.05 sin x superimposed over a scatter diagram of the data. Find the sinusoidal function of the form y = A sin ( x - ) + B w hich best fits the data. π π ππ ] + 66.252 45.05sin[ A) yπ= β 45.05 B) y = 45.05 sin t + 12 + 21.20 π sin 6 x - 3 + 66.25 8 C) y = 45.05 sin 4 x- 2 3 + 21.20 D) y = 45.05 sin Page 584 6 x- 12 + 66.25