Checkpoint 1 Define Trig Functions Solve each right triangle by finding all missing sides and angles, round to four decimal places. 1.. B P 10 8 Q R A C. Find the measure of A and the length of side a.. A surveyor is 980 feet from the base of the world s tallest fountain at Fountain Hills, Arizona. The angle of elevation to the top of the column of water is 9.7 o. His measuring device is at the same level as the base of the fountain. Find the distance from the measuring device to the top of the fountain. 5. A weather balloon is 150 meters away from a meteorologist. If the angle of depression is 1. o, then find the height of the weather balloon.. An airplane has an altitude of,000 feet and is 1 miles away from its destination, which is at sea level. Find the angle of elevation from the destination to the airplane. 7. Luca rode a horse 5 meters down a straight slope from the top to the bottom of a coneshaped hill. The slope made a 17 degree angle to the horizontal baseline under the hill. How many meters in height was the hill?
8. Gianna stands 5 meters from a tree. Using a laser range finder (which uses a beam of light to find the distance between objects), she determines her feet are 50 meters from the top of the tree. How many meters tall is the tree? 9. David uses scissors to cut a rectangular piece of paper from one corner to the opposite corner. The paper is 1 centimeters long (from the corner where he started the cut) and the cut makes a 5 degree angle to the uncut edge. How many centimeters long is the cut? 10. Zoe stood 1 meters from a tree. She placed a straight stick on the ground at her feet, angled it up until it pointed to the top of the tree, and measured the angle as degrees. How many meters tall was the tree? 11. Lisa carries a box up a metal ramp into the back of a truck. The ramp is meters long and the bottom makes a 5 degree angle with the ground. How many meters is the ground distance between the bottom of the ramp and the back of the truck? 1. Alex is standing on the diving board at the local pool. Two of her friends are in the water on the opposite side of the pool. If the angle of depression to one of her friends is 0 degrees, and 0 degrees to her other friend who is 5 feet farther away than the first friend, then how tall is the diving platform? Given the following function, find the other 5 trig functions: 5 7 1. sin θ = 1. tan θ = 15. sec θ =
Checkpoint Special Right Triangles Use Special Right Triangle Ratios to solve for x and/or y in the following examples. 1.... 5.. 7. 8.
9. Find x, y, and z and the perimeter of Trapezoid PQRS 10. Owen is building a bean bag toss for the school carnival. He is using a -foot back support that is perpendicular to the ground feet from the front of the board. He also wants to use a support that is perpendicular to the board as shown in the diagram. How long should he make the support? 11. Suppose a zip line is anchored in one corner of a course shaped like a rectangular prism. The other end is anchored in the opposite corner as shown. If the zip line makes a 0 angle with the post AF, find the zip line s length, AD. Find the requested ratios without using a calculator. 1. sin 0 1. cot 0 1. sec 0 15. tan 0
Checkpoint Drawing Angles Co-terminal Angles Draw an angle with the given measure in standard position: 1. 0. -15. -05. 750 Find one positive and one negative angle that is co-terminal with 5. 0. -150 7. 570 Draw an angle with the given measure in standard position, then locate and name its reference angle: 8. 150 9. 0
10. 0 11. 0 Write an expression that describes all co-terminal angles for the following: 1. 5 1. -77 1. 15. Find a co-terminal angle within one revolution for each of the following: 15.,5 1. -177, 89 Find the requested ratios without using a calculator. Then, write the co-function for each. 17. csc 0 o 18. sin 0 o 19. cot 5 Isabelle and Karl agreed to meet at the rotating restaurant at the top of a tower in the town center. The restaurant makes one full rotation each hour. 0. Isabelle waits for Karl, who arrives 0 minutes later. Through how many degrees does the restaurant rotate between the time that Isabelle arrives and the time that Karl arrives? 1. Since the restaurant is busy, they don t get menus for another ten minutes. How much farther has the restaurant rotated?. By the time they are served dinner, the restaurant has rotated to an angle 0 short of its orientation when Isabelle arrived. a. How long has she been there? b. How long has Karl been there? c. How far has the restaurant rotated since Karl arrived?. When their bill comes, it includes a note that the restaurant has rotated 80 since Isabelle arrived. a. How long has it been since Isabelle arrived? b. How many rotations has the restaurant made since Isabelle arrived?
Checkpoint - Defining Radian Measures Draw an angle with the given measure in standard position: 1... 5. 7 5.. 5 7. 8.
Convert the following radians to degrees. 9. 10. 5 11. 17 1. 5 1. 1.. Convert the following degrees to radians. 11. 70 1. 510 1. 0 1. 15 15. -180 1. -5 17. How many radians are there in the 0 degrees? Leave in terms of π and round to two decimal places. 18. The minute hand of a clock travels how many radians in 15 minutes? How about 0 minutes? Leave in terms of π and round to two decimal places. 19. Suppose you travel around the circle, what would your reference angle be in radians? In degrees? 0. Suppose you travel around a circle, what would be the length you actually traveled given that the radius of the circle is 9 centimeters, and the formula for circumference is C r?
Important radians to have memorized. Checkpoint 5 - Defining Radian Measures Find one positive and one negative angle that is coterminal with 1.. 5.. 7 Draw an angle with the given measure in standard position: 5. 5. 1 Trigonometric functions with degrees and radians. Find the requested ratios without using a calculator. 7. cos π 8. tan 5π 9. sin 11π 10. csc 7π 11. cot 1. sin 1. sec 5 1. tan 5
15. Which of the following angles have a reference angle with a measure of 0? I. θ = 10 II. θ = 10 III. θ = 0 A. III only C. II and III only B. I and II only D. I, II, and III 1. For what values of θ in the interval [0, π) does the cos θ have the same value as sin π? A. cos π and cos 5π C. cos π and cos 5π B. cos π and cos 7π D. cos π and cos 7π 17. For what values of θ in the interval [0, π) does the sin θ have the same value as cos π? A. sin π and sin 5π C. sin 5π and sin 7π B. sin π and sin 7π D. sin π and sin 7π 18. The reference angle for 11π is A. 7π B. π C. π D. 5π E. none
Checkpoint Deriving the Unit Circle Using your unit circle, fill in the points in Quadrant I that represent the cosine and sine values. 1) 0 o or Triangle 5 o or Triangle 0 o or Triangle ) Using your Quadrant I diagram, fill in the following tangent values for π, π, and π. tanθ = sinθ cosθ. a) tan π b) tan π c) tan π Find the value of each by using your points (1,0), (0,1), (-1,0), and (0,-1) from your unit circle. ) sin0 ) cos0 5) tan0 ) cos π 7) sin π 8) tan π 9) cos π 10) sin π 11) tan π 1) cos π
1) Practice filling in this unit circle with the radian values for each given angle: For example, in the first box, write 0 and/or π and in the second box, write π, etc. 90 180 0 70 1) A unit circle is drawn below, where θ = π Find the exact value of sin θ. and Point R is the midpoint of OS. A. 1 C. B. 1 D.
Checkpoint 7 Using the Unit Circle For #1 1, find the requested value without using a calculator. Simplify radical answers. 1. sin 5 o. tan 15 o. csc 70 o. cos 150 o 5. cot. cos 7. 5 sec 8. 11 sin 9. sin 10 o 10. tan 0 o 11. csc 1. cos 0 o 1. cot 1. cos 10 o 15. sec 1. 5 sin For #17, find all possible values (degrees and radians) for the missing angle in the equation: 17. sin 18. tan 1 19. csc 0. 1 cos 1. cot undefined. cos. sec. sin
Checkpoint 8 Arc Length Find the radian measure of the central angle of a circle of radius r that intercepts an arc length of s. 1. r = 8 inches s = inches. r =.5 meters s = 800 cm Find the degree measure (nearest degree) of the central angle of a circle of radius r that intercepts an arc length of s.. r = 117 yards, s = 5 yards Given the following radius and central angle, find the measure of the intercepted arc.. r = 5 ft, θ = π 5. r = 107 inches, θ = 0. r = 1 cm, θ =.1 7. John is adding a curved edge to the landscaping in front of the high school. The curve is an arc of a circle with a radius of 100 feet. The central angle that intercepts the curve measures radians. Find the length of the curve to the nearest foot.
8) Which of the following angles have a reference angle with a measure of 0? I. θ = 10 II. θ = 10 III. θ = 00 A. III only C. II and III only B. I and II only D. None of these 9) What point on the unit circle does the terminal side of an angle of 7π pass through? A. B. (, ) C. (, 1 ) D. (, ) (, 1 ) 10) If 0 θ π and sin θ =, then which of the following are possible values of θ? I. II. III. IV. π π 5π 7π A. I and IV C. III and IV B. III only D. II and III State in which quadrant the terminal side of the angle lies. 11).5 1). 1) 7π 8 1) 17π 15),7 1).75
Checkpoint 9 Angular Velocity Convert each from revolutions per minute (rpm) to radians per second. Write your answer in terms of π and in decimal form (nearest tenth). 1) 5 rpm ) rpm Convert each from radians per second to revolutions per hour. Round to the nearest tenth. ).5 radians/sec ) π 7 radians/sec 5) An object is rotating at 71,58. radians per hour. Find the number of revolutions it is rotating per minute. Round to the nearest tenth. ) A point is rotating at,5π radians per hour. Find the number of revolutions it is rotating per second. Round to the nearest tenth. 7) A DVD rotates through an angle of 0π radians per second. At this speed, how many revolutions does the DVD make while playing a movie clip that is minutes and 15 seconds long? A. 900 revolutions C. 0.5 revolutions B. 1950 revolutions D..5 revolutions
8) Tommy on the Ferris Wheel: Tommy is riding on a Ferris wheel with a radius of 0 feet. The wheel is rotating at 1.5 revolutions per minute (nonstop). Find Tommy s angular velocity in radians per second. Write your answer in terms of π and in decimal form to the nearest hundredth. Find the positive radian measure of the angle that the second hand of a clock moves through in the given time. Leave in terms of π. 9) minutes and 0 seconds 10) minutes and 15 seconds 11) The Earth makes one rotation every hours. How many radians per hour is this? Leave your answer in terms of π. Review Questions 1. Which of the following angles has a reference angle with a measure of 0? I. θ = 10 II. θ = 150 III. θ = 0 A. III only C. II and III only B. I and II only D. I, II, and III 1. What point on the unit circle does the terminal side of an angle of 11π pass through? A. (, ) C. (, ) B. (, 1 ) D. (, 1 ) 1. Find the exact value of csc 5π. A. C. B. D.
Checkpoint 10 Radian Practice Change each to radian measure. Leave your answer in terms of π 1) 0 0 ) -150 0 Change each to degree measure. ) 1π/5 ) 5.7 Find the arc length of a circle when given the central angle and the radius of the circle. Leave answer in terms of π and round to the nearest tenth. 5) π/ with radius = 8 feet ) 15 0 with radius = 8 feet 7) Find the radian measure of a central angle in a circle with radius 1 feet and an arc length of 0 feet. 8) Find the degree measure of a central angle (to the nearest tenth) in a circle with radius inches and an arc length of 1 inches. 9) Find the angular velocity of an object rotating.5 revolutions per minute. Leave in terms of π 10) Through what radian measure does a second hand travel in minutes and 0 seconds? Leave in terms of π
Find a positive co-terminal angle within one revolution for each. 11) 50 0 1) 1, 0 1) -1π/5 1) 7.8 Find the reference angle for each. 15) 710 0 1) 5π/ 17) 1π/5 18) -5π/ On this next review section, do not use a calculator. Find the exact value for each. 19) sec(π/) 0) cot(5π/) 1) sin(-0 0 ) ) csc(7π/) Find a co-function for each. ) sin(0 0 ) ) tan(5 0 ) 5) sec(π/7) Use a calculator for these. Solve for A. Round to the nearest tenth of a degree. ) sina = 5/7 7) cosa =.1889 8) tana = 1/7
Checkpoint 11 Deriving Trig Identities Simplify each of the following trigonometric expressions. Write the three Pythagorean Identities, along with the three forms of each. Simplify each of the following trigonometric expressions. 1) 15(cos x + sin x) ) 17sec x 17tan x ) cos (70 ) +cos (0 ) ) cot (x) csc x 5) cos (x) + sin (x) ) + tan x Write the three reciprocal identities. secx = cscx = cotx = Use the above identities in order to simplify the following expressions. 7) (cosx)(sinx)(secx)(cscx) 8) sec x + csc x
9) (cotx)(5sinx)(tanx)(sinx) 10) 9 tan x + 9 11) (tan x + sec x 1)(cot x) 1) (1 cos x)(csc x) Solve each equation for x where 0 x < π. You must write your answer in radians! 1) sinx = 0 1) cosx = -1 15) secx = - 1) tanx = 0 17) tanx = -1 18) sinx = 1 19) cosx = 0) cotx = - 1) sinx = - ) If sinx = -1 then what is cosx? ) If cosx = then what are the possible values of sinx?
Checkpoint 1 Graphing Sine/Cosine Functions Graph the following trig functions using a table, and then state the domain and range. 1. y sin x. f ( x) cos x Domain: Range: Domain: Range: x y x y. y cos x. f ( x) sin x Domain: Range: Domain: Range: x y x y
x 5. y cos. f ( x) sin x Domain: Range: Domain: Range: x y x y Find the requested ratios without using a calculator. 7. sin 8. 17 cos 9. 9 sec 10. tan 1 11. csc 1. 11 sin 1. 5 tan 1. cos Solve each equation for x where 0 x < π. You must write your answer in radians! 15. 1 cos x 1. sin x 0 17. tan x
Checkpoint 1 Graphing Sine/Cosine/Tangent Functions Graph the following trig functions using a table, and then state the domain and range. 1. y sin x. f x x ( ) cos 1 Domain: Range: Domain: Range:. y cos x. f ( x) sin x Domain: Range: Domain: Range:
tan. f ( x) tan x 5. y x Domain: Range: Domain: Range: x y x y 7. y sin x 8. f x x ( ) tan Domain: Range: Domain: Range:
Find the requested ratios without using a calculator. 9. 7 sin 10. cos 11. 5 sec 1. 19 tan 1. 7 csc 1. 7 sin 15. tan 1. 5 cos Solve each equation for x where 0 x < π. You must write your answer in radians! 15. cos x 1 1. sin x 17. tan x 1