Mathematics IV
1st Edition
Unit 4
Name: ________________________
More Relationships in the Unit Circle Learning Task:
1. In Math II, you learned three trigonometric ratios in relation to right triangles. What are these
relationships?
hyp otenuse
opp osite

adjacent
2. There are three additional trigonometric ratios that you will use in this unit: secant, cosecant and
cotangent.
hypotenuse
sec  
adjacent
hypotenuse
csc  
hyp otenuse
opp osite
opposite
adjacent
cot  

opposite
adjacent
How do these ratios relate to the trigonometric ratios from #1?
3. Moving the triangle onto the unit circle allows us to represent these six trigonometric relationships in
terms of x and y. Express each of the six ratios in terms of x and y.
(x, y)
1

4. Based on these relationships x = _____________ and y = _____________. This is a special case of the
general trigonometric coefficients (rcos, rsin) where r = 1.
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Unit 4: Page 1 of 3
Mathematics IV
5.
Unit 4
1st Edition
a. Use this relationship to determine the coordinates
of A. Both coordinates a positive. Why is this
true?
b. What angle would have (-0.9397, -0.3420) as its
coordinates on the unit circle? Why?
c. What angle would have (0.9397, -0.3420) as its
coordinates? Why?
6.
a. What is the reference angle for 250o?
b. What are the coordinates of this angle on the
unit circle?
c. What 2nd quadrant angle has the same reference
angle? What are the coordinates of this angle
on the unit circle?
7. Using a scientific or graphing calculator, you can quite easily find the sine, cosine and tangent
of a given angle. This is not true for secant, cosecant, or cotangent. Remember from Math II ,
1
that sin-1θ is not the same as
. Since the three new trigonometric ratios are not on a
sin 
calculator, how can you use the definitions of the ratios from #2 to calculate the values?
8. A student entered sin 30 in her calculator and got -0.98803. What went wrong?
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Unit 4: Page 2 of 3
Mathematics IV
1st Edition
Unit 4
9. Based on the graph of the unit circle on the grid, estimate each of the values. Do not use the trig
keys on the calculator for this problem. You will need to use a protractor to mark each angle
and then estimate the coordinates where the terminal side of the angle intersects the unit circle.
a. sec 60o
d. sec -75o
b. csc 180o
e. csc 490o
c. cot 235o
f. cot 920o
10. Use a calculator to find each of the following values.
a. sin 40o
e. tan 300o
b. csc 40o
f. cot 300o
c. cos 165o
g. csc 90o
d. sec 165o
h. sec -140o
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Copyright 2010 © All Rights Reserved
Unit 4: Page 3 of 3
Mathematics IV
Unit 4
1st Edition
More Relationships in the Unit Circle Learning Task:
1. In Math II, you learned three trigonometric ratios in relation to right triangles. What are these
relationships?
Solution
opposite
hypotenuse
adjacent
cos  
hypotenuse
opposite
tan  
adjacent
sin  
hyp otenuse
opp osite

adjacent
2. There are three additional trigonometric ratios that you will use in this unit: secant, cosecant and
cotangent.
hypotenuse
sec  
adjacent
hypotenuse
hyp otenuse
csc  
opp osite
opposite
adjacent
cot  

opposite
adjacent
How do these ratios relate to the trigonometric ratios from #1?
These ratios are the reciprocals of the ratios in #1.
3. Moving the triangle onto the unit circle allows us to represent these six trigonometric relationships in
terms of x and y. Express each of the six ratios in terms of x and y.
(x, y)
1

y
1
x

1
x

y
1

y
1

x
y

x
sin  
cos 
tan 
csc 
sec 
cot 
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Copyright 2010 © All Rights Reserved
Unit 4: Page 4 of 3
Mathematics IV
Unit 4
1st Edition
4. Based on these relationships x = _____________ and y = _____________. This is a special case of the
general trigonometric coefficients (rcosθ, rsinθ) where r = 1.
x = cosθ
5.
y=sinθ
a. Use this relationship to determine the
coordinates of A. Both coordinates are positive. Why is
this true?
(0.9397, 0.3420) The angle is in Quadrant I.
b. What angle would have coordinates
(-0.9397, -0.3420) on the unit circle?
Why?
200o - The reference angle will be 20o and the angle
will be in Quadrant III. 180o + 20o = 200o
c. What angle would have (0.9397, -0.3420)
as its coordinates? Why?
340o - This is a fourth quadrant angle with a
reference angle of 20o.
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Copyright 2010 © All Rights Reserved
Unit 4: Page 5 of 3
Mathematics IV
6.
Unit 4
1st Edition
a. What is the reference angle for 250o?
70o
b. What are the coordinates of this angle on the unit circle?
(-0.3420, -0.9397)
c. What 2nd quadrant angle has the same reference angle?
What are the coordinates of this angle on the unit circle?
110o; (-0.3420, 0.9397)
7. Using a scientific or graphing calculator, you can quite easily find the sine, cosine and tangent of a given
angle. This is not true for secant, cosecant, or cotangent. Remember from Math II, that sin-1 is not the
1
same as
. Since the three new trigonometric ratios are not on a calculator, how can you use the
sin 
definitions of the ratios from #2 to calculate the values?
To find the value of the cosecant of an angle, first find the sine of the angle and then take the
reciprocal. To find the secant of an angle, find the reciprocal of the cosine value. To find the
cotangent value, find the reciprocal of the tangent.
8. A student entered sin30 in her calculator and got -0.98803. What went wrong?
The calculator was in the wrong mode. The mode should be degrees. (Note: This is a good time
to explain that angles can be measured in degrees or radians, which will be introduced in an
upcoming task.)
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Copyright 2010 © All Rights Reserved
Unit 4: Page 6 of 3
Mathematics IV
1st Edition
Unit 4
9. Based on the graph of the unit circle on the grid, estimate each of the values. Do not use the trig keys on
the calculator for this problem. You will need to use a protractor to mark each angle and then estimate
the coordinates where the terminal side of the angle intersects the unit circle.
Solutions:
a. sec 60o
b. csc 180o
c. cot 235o
1
2
.5
1
 undefined
0
 .5736
 .7002
 .8192
d. sec -75o
e. csc 490o
f. cot 920o
10. Use a calculator to find each of the following values.
a. sin 40o
0.6428
e. tan 300o
b. csc 40o
1.5557
f. cot 300o
o
c. cos 165 -0.9659
g. csc 90o
d. sec 165o -1.0353
h. sec -140o
1
 3.864
.2588
1
 1.3055
.7660
 .9397
 2.7477
 .3420
-1.7321
-0.5774
1
-1.3054
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Unit 4: Page 7 of 3