Download More Relationships in the Unit Circle Learning Task:

NAME _____________________________
More Relationships in the Unit Circle Learning Task:
1. In MII, you learned three trigonometric ratios in relation to right triangles. Name these relationships.
sin  
hyp otenuse
opp osite
cos  

adjacent
tan  
2. There are three additional trigonometric ratios that you will use in this unit: secant, cosecant and
cotangent.
hypotenuse
opposite
hypotenuse
(secant ) sec  
adjacent
adjacent
(cotangent ) cot  
opposite
(cosecant ) csc  
hyp otenuse
opp osite

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

NAME _____________________________
4. If I were to solve for x using the cos identity above, what would x = ? __________
5. If I were to solve for y using the sin identity above, what would y = ? ___________ This is a
special case of the general trigonometric coefficients (rcos, rsin) where r = 1.
6.
a. Use the relationship from # 4 to determine the
coordinates of A. (decimal form)
______________________
Both coordinates are positive. Why is this true?
______________________________________
b. What angle would have coordinates (-0.9397, -0.3420)
on the unit circle? ______________________
Why? ____________________________________
c. What angle would have (0.9397, -0.3420) as its
coordinates? _________________________
Why? __________________________________
7.
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? __________________________
NAME _____________________________
8. 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 MII, that sin-1 is
1
not the same as
. Since the three new trigonometric ratios are not on a calculator, how
sin 
can you use the definitions of the ratios from #2 to calculate the values?
_____________________________________________________________________
9. Use a calculator to find each of the following values. Round answers four decimal
places.
a. sin 40o
e. tan 300o
b. csc 40o
f. cot 300o
c. cos 165o
g. csc 90o
d. sec 165o
h. sec -140o
10. Given that an angle has a point on the unit circle at
, find the six trigonometric function values
for that angle. SHOW YOUR REASONING!
sin   _______
csc   _______
cos   _______
sec   _______
tan   ________
cot   _______
11. Given that the point (-3, -7) lies on the terminal side of an angle , find the 6 trigonometric functions.
sin   _______
csc   _______
cos   _______
sec   _______
tan   ________
cot   _______