Question 1

A forester, 200 feet from the base of a redwood tree, observes that the angle between the ground and the top of the tree is  . Estimate the height of the tree.

Accepted Answer

A) 386.4 feet 
B) 22.6 feet 
C) 376.1 feet 
D) 367.5 feet 
E) 219.5 feet

Question 2

Express the angle as a decimal, to the nearest ten-thousandth of a degree.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 3

Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates  , find  ,  ,  ,  .

Accepted Answer

A)  
 
B)  
 
C)  
 
D)  
 
E)

Question 4

If a circular arc of the length s = 13.5 cm subtends the central angle  = 3 on a circle, find the radius of the circle.

Accepted Answer

A) 4.5 cm 
B) 5.5 cm 
C) 6 cm 
D) 40.5 cm 
E) 5 cm

Question 5

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation  while the relative humidity in percent could be expressed by  where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

Accepted Answer

A) H = 85% 
B) H = 10% 
C) H = 55% 
D) H = 90% 
E) H = 35%

Question 6

On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 400 calories/cm 2  at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 7

A ship leaves port at 4:00 P.M. and sails in the direction  at a rate of 19 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction  at a rate of 16 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?

Accepted Answer

A) 70 mi 
B) 17 mi 
C) 75 mi 
D) 25 mi 
E) 140 mi

Question 8

A builder wishes to construct a ramp 23 feet long that rises to a height of 4.5 feet above level ground. Approximate the angle that the ramp should make with the horizontal. Round the answer to the nearest degree.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 9

Find the exact value.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 10

A builder wishes to construct a ramp 22 feet long that rises to a height of 5.5 feet above level ground. Approximate the angle that the ramp should make with the horizontal. Round the answer to the nearest degree.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 11

As  for some real number  . Use a graph to predict  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 12

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation  while the relative humidity in percent could be expressed by  where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

Accepted Answer

A) H = 10% 
B) H = 90% 
C) H = 35% 
D) H = 55% 
E) H = 85%

Question 13

On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 480 calories/cm 2  at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 14

The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by  with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 19 days.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 15

Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60 o , what is the new location of the hand?

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 16

Refer to the graph of y = cos x to find the separate values of x in the interval  that satisfy the equation.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 17

Sketch the graph of the equation.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 18

Scientists sometimes use the formula  to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in  , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 34  occurs at 2 P.M., and the average temperature of 19  occurs 6 hours later.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 19

As  for some real number  . Use a graph to predict  .

Accepted Answer

A)   
B)   
C)   
D)   
E)

Question 20

Simplify the expression.

Accepted Answer

A)   
B)   
C)   
D)   
E)

Exam 6: The Trigonometric Functions

A forester, 200 feet from the base of a redwood tree, observes that the angle between the ground and the top of the tree is . Estimate the height of the tree.

Express the angle as a decimal, to the nearest ten-thousandth of a degree.

Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , .

If a circular arc of the length s = 13.5 cm subtends the central angle = 3 on a circle, find the radius of the circle.

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation while the relative humidity in percent could be expressed by where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 400 calories/cm² at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.

A ship leaves port at 4:00 P.M. and sails in the direction at a rate of 19 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 16 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?

A builder wishes to construct a ramp 23 feet long that rises to a height of 4.5 feet above level ground. Approximate the angle that the ramp should make with the horizontal. Round the answer to the nearest degree.

Find the exact value.

A builder wishes to construct a ramp 22 feet long that rises to a height of 5.5 feet above level ground. Approximate the angle that the ramp should make with the horizontal. Round the answer to the nearest degree.

As for some real number . Use a graph to predict .

On March 17, 1981, in Tucson, Arizona, the temperature in degrees Fahrenheit could be described by the equation while the relative humidity in percent could be expressed by where t is in hours and t = 0 corresponds to 6 AM. Find the maximum value of the humidity.

On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 480 calories/cm² at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.

Refer to the graph of y = cos x to find the separate values of x in the interval that satisfy the equation.

Sketch the graph of the equation.

As for some real number . Use a graph to predict .

Simplify the expression.