Trig Review: radians, 6 trig ratios, special right triangles, unit circle
Write down how you would describe radian measurement.
In your group, share your individual descriptions of a radian. Ask questions of each other if explanations are
unclear or lacking. Make any additions or edits to your description.
Determine if the following statements are true or false. If false, explain why. Make sure everyone in the
group agrees and understands why.
1.
One radian is always the same unit length.
2.
One radian is always the same arc length.
3. If the radius of a circle doubles, then the arc length (associated with a fixed central angle) doubles.
4. If the arc length of a circle doubles, then the radius is doubled and the central angle is doubled.
5. If the central angle of a sector doubles and there is no change to the arc length, then the radius has also
doubled.
How many radii does it take to go around the circumference of a circle? Does the size of the circle matter?
Fill in the missing values in the table.
Central angle
(in radians)
5
2
Exact arc length
10
3
10 cm
Radius length
2 cm
7 cm
14 cm
3
cm
2
6 cm
6. A railroad curve is to be laid out on a circle. What radius should be used to change direction by 25 if 120
feet of track is used?
7. A freight train is moving at the rate of 8 miles per hour along a piece of circular track of radius 2,500 feet.
Through what angle (in radians) does it turn in 1 minute?
8. A low earth orbit satellite is in an approximately circular orbit 300 km above the surface of the Earth. If the

ground station tracks the satellite when it is within a radian central angle above the tracking antenna (directly
4
overhead), how many kilometers does the satellite cover during the tracking? Assume the radius of the Earth is
6400 km. Round to the nearest kilometers.
9. The famous clock tower in London has a minute hand that is 14 ft. long. How many feet does the tip of the
minute hand of Big Ben travel in 25 minutes? How long does it take to cover a mile?
10. The planet Jupiter rotates every 9.9 hours and has a diameter of 88,846 miles. If you are standing on the
equator, how fast are you traveling (linear speed)?
11. If a contestant on The Price Is Right lands on the $1.00 space on his first spin, he is granted a second spin.
If this spin lands on either the $0.05 or $0.15 space, he wins $5,000; if it lands on the $1.00 space again, he
wins $10,000. This time, he tries to control the spin so that the wheel makes exactly one complete revolution,
84
 ft, does the contestant win any money? Justify
again landing on $1.00. If the arc length of his next spin is
11
your answer.
12. NASA explores artificial gravity as a way to counter the
physiologic effects of extended weightlessness for future space
exploration. NASA’s centrifuge has a 58-foot-diameter arm. If two
humans are on opposite end of the centrifuge and they rotate one full
rotation every second, what is their linear speed in feet per second?
13. The large gear has a radius of 6 cm, the medium gear has a radius of 3 cm, and the small gear has a radius
of 1 cm. If the small gear rotates 1 revolution per second, what is the linear speed of a point traveling along the
circumference of the large gear? Is it faster, slower or equal to the linear
speed of a point on the middle gear?
Determine the six trig ratios (in terms of x, y, r) for the given angle  .
sin   _______
cos   _______
tan   ______
csc   _______
sec   _______
cot   ______
Which trig ratios have range values between 1 and -1? ____________________________________________
Which trig ratios have a range of all real numbers? _____________________________________
Which trig ratios have range values < -1 or range values > 1? _____________________________
Sketch a graph of the two special right triangles. Remember to describe  in radians.
Fill in the unit
circle.
14. Find the values of the trig functions of the acute angle A of the right triangle ABC given a  2, c  2 5
15. When the sun is

9
radians above the horizon, how long is the shadow cast by a building 150 feet high?
16. Find the height of a tree if the angle of elevation of its top changes from

9
radians to
2
radians as the
9
observer advances 75 feet toward the base of the tree.
17. A tower standing on level ground is due north of point N and due west of point W, a distance 1,500 feet
from N. If the angles of elevation of the top of the tower as measured from N and W are
4
11
radians and
radians , respectively, find the height (to the nearest foot) of the tower.
90
180
18. The red stripe on a barber pole makes two complete revolutions around the pole. If the pole is 1 meter high
and 16 meters in diameter, what angle does the stripe make with the horizontal? What is the length of the
stripe?
19. Find the value of the six trig functions for the angle in standard position through the point (-3, 5).
20. Find (without using a calculator) the exact value of each expression.
 7
a) sin 
 6

 11
  cos 

 6

 11 
 7
 tan 
 b) cos 

 6 
 4

 4
 tan 

 3

 7 
  cos 


 6 
 5
c) csc 
 3

 7
  tan 

 6

 3 
  sin 


 4 
21. Find the value of each expression (without using a calculator).
a) cot   1,

2
  ;
find cos 
c) cos   0.5 and tan   3;
find csc 
2 3 3
,
   2 ; find sin 
3
2
2 3
1
d )sec  
and sin    ; find cot 
3
2
b) sec  
22. Find all the values of  , 0    2 , for which the equation sin   cos is true.
23. Does there exist an angle 0    2 such that tan   cot  ?
24. Does there exist an angle 0    2 such that sec   csc( ) ?
Explain the mistake that is made.
 5 
25. Evaluate tan 
 exactly.
 6 
 5 
sin 

6 
 5 

tan 

 6  cos  5 


 6 
 5
sin 
 6
3

 5  1
and cos 


2

 6  2
 5 
tan 

 6 

3
2
1
2
 5 
tan 
 3
 6 
This is incorrect.
 11
26. Evaluate sec 
 6
 11
sec 
 6

 exactly.

1


 cos  11 


 6 
 11
cos 
 6
1


2

1
 11 
sec 
 1
 6  
2
 11 
sec 
  2
 6 
This is incorrect.