Download Intro to the Unit Circle

Trigonometry
Mr. M’s Mad Math Moment
Would like to now introduce (drum roll please);
The Unit Circle
Background Info
y
You studied three ratios of triangles and then introduced to their ratios when
placed on the coordinate plane.
sin  
P (x,y)
r=1
y
x
y
, cos 
, tan  
r
r
x
There are three reciprocal ratios;
csc 

y
x
r
x
r
, sec 
, cot 
y
y
x
If you use a circle who’s radius = 1 unit , then you utilize something called
“The Unit Circle”. Locate the terminal side of any angle formed and then
find its inter section with “The Unit Circle”. This location will determine
the value of the trig function.
x
Pythagorean Theorem
x2  y2  r 2
Decimal approximations of many numbers are not exact, no matter how many decimal places are used. For example
4/7 is written as 0.5714 as a decimal but is not an exact value. For some formulas and calculations it is important to
know the exact value rather than a decimal approximation. In this activity, you will build your knowledge base and
develop values for certain special angles.
Assignment
Start by completing the angles on the unit circle that are “multiples” of the specified angle on
the appendix page provided.
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Introducing the Unit Circle
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revised: 3/20/2014
Background Info
You studied three ratios of triangles called the Sine, Cosine, and
Tangent. You were then introduced to three reciprocal ratios
called the Cosecant, Secant, and Cotangent. What you are going
to do now is use the unit circle to learn about the exact values of
these six trig functions for some special angles.
sin  
y
r
,
cos 
x
r
,
tan  
y
P (x,y)
r=1
y
x
y
There are three reciprocal ratios;
a=1

r
x
r
csc  , sec  , cot 
y
y
x
O
x
x
Q (1,0)
x
Pythagorean Theorem
x2  y2  r 2
Discovery
Assume that an angle () is rotated from the positive x-axis counter-clockwise until it terminates at 60°. Determine
what the x & y coordinates are on the point on the unit circle, by considering a 30°, 60°, 90° triangle formed. It
might also help if you mirror that triangle to create an equilateral triangle (OPQ ) as shown above, that way
x = ½ because 2x = 1 in the equilateral triangle that is formed.



Use the Pythagorean theorem to find the y coordinate of point P. in “EXACT” radical form (no decimal
numbers – no calculators).
With your knowledge of the unit circle, what is the “sine” of a 60° angle. (in other words sin60° = ?)
With your knowledge of the unit circle, what is the “cosine” of a 60° angle. (in other words cos60° = ?)
Now, assume the terminal side of the angle intersects the “unit circle” at the point H(x,y). If the
terminal side rotates to 45°, what would the x & y coordinates of the point H be ?
Use what you know about a 45°-45°-90° triangle and the Pythagorean Theorem to find the
exact (meaning in radical, not decimal form) value of the trig functions. Leave your answers in
simplified, but improper fractional form.
After you do this work – Mr. M will lead a class discussion
looking at what you found out.
(he’ll also make sure you did your algebra correctly)
Introducing the Unit Circle
2
revised: 3/20/2014
Appendix
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Introducing the Unit Circle
3
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revised: 3/20/2014
In the unit circle, If = 60° x = ½ , because 2x = 1 in the equilateral triangle that is
formed. In the space to the right, use the Pythagorean theorem to compute the ycoordinate, then fill in the blanks with what you know.
y
P (x,y)
r=1
a=1
y
 = 60°

O
Q (1,0)
x
x
x
X = ________
Y = ________
Radius = _______
In the unit circle, If = 45°, what do you know about the x & the y-coordinates in
the triangle formed? In the space to the right, use the Pythagorean theorem to
compute the x-coordinate & the y-coordinate then fill in the blanks with what you
know. Think about 45°-45°-90° triangles.
y
H (x,y)
r=1
y

O
x
 = 45°
Q (1,0)
x
X = ________
Y = ________
Radius = _______
Introducing the Unit Circle
4
revised: 3/20/2014