Download MA 15800 Lesson 20 Notes Summer 2016 The Unit Circle

MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
Thus far, we have defined trigonometric functions (sine, cosine, tangent, cosecant, secant, and
cotangent functions) as functions of angles in either degree measurements or radian
measurements. In calculus and many other applied fields, the domains of trigonometric
functions are real numbers or a subset of real numbers. The notation cos 5 could be
interpreted as cosine of 5 radians or the cosine of the real number 5. (It would not be interpreted
as cosine of 5 degrees unless there was a degree symbol.)
All six trigonometric functions may be defined as functions of a real number t rather than as a
function of an angle (as explained above). Let a point P be an ordered pair 𝑃(𝑥, 𝑦) on a unit
circle. (A unit circle is a circle whose center is at the origin of a coordinate system and whose
radius is 1 unit.) This point corresponds to the real number t.
Let’s discuss another way to derive the trigonometric functions, the unit circle. Draw a
coordinate plane and only mark one unit in each direction.. Draw a circle that is centered at the
origin and has a radius of one. This is the unit circle. Mark your x-axis and y-axis and also write
cosine next to the x-axis and sine next the y-axis.
y-axis
sine θ
1
x-axis
cosine
θ
Draw a right triangle with a central angle,  = 60,
and mark point P on the circle. The x-value of the
coordinate is equal to the cos(60) and the
y-value of the coordinate is equal to sin(60)
cos(60) =
y-axis
sine
P  1 ,
3
 2 2 


60°
1
x-axis
cosine
1
3
and the sin(60) =
2
2
If we draw the corresponding right triangle for a 45 central angle, the corresponding point is
 1 1 
1
1
P , , therefore cos(45) =
and sin(45) =
y-axis
 2 2 
2
2
sine
1
1
P
,

2
2

45°
1
x-axis
cosine
1
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
Draw the corresponding right triangle for a 30 central angle and the corresponding point is
 3 1 
1
3
y-axis
and the sin(30) =
P
 2 , 2 
, therefore cos(30) =
2
2


sine
3
sin60
3 2
sin
 2 
  3,
Since tangent =
then; tan(60) =
cos60 1
2 1
cos
2
1
1
sin30
1 2
1
3
sin(45)
2
 2  


tan(30)=
and, tan(45)=

1
1
cos30
3 2 3
3
3
cos(45)
2
2
P
30°
1
x-axis
cosine
This discussion of using points on a unit circle to define the trigonometric
functions leads to defining the trigonometric functions as functions of real
numbers, not just functions of angles. Let t be a real number that
corresponds to the point P on the unit circle as seen below.
Then the six trigonometric functions are defined as below where t is a real
number corresponding to the point, 𝑷(𝒕) = (𝒙, 𝒚).
sin t  y
csc t 
1
,y0
y
cos t  x
1
sec t  , x  0
x
y
, x0
x
x
cot t  , y  0
y
tan t 
2
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
Ex 1: Find the six trig values given the point below on a unit circle.

3 4
5 5
P  ,

sin t 
cos t 
tan t 
csc t 
sec t 
cot t 
Ex 2: Find the six trigonometric values for the point on a unit circle given below.
R

1
2
,
1
2

sin t 
cos t 
tan t 
csc t 
sec t 
cot t 
Ex 3: Find the six trigonometric values (if they exist) for the point on a unit circle given below.
S (0,1)
sin t 
cos t 
tan t 
csc t 
sec t 
cot t 
Note: The point for example 3 above corresponds to an angle of 90°. It is easy to
determine the trig values for the real numbers associated with the angles of 0°, 90°, 180°,
and 270°. The points are (1,0), (0,1), (-1,0), and (0, -1).
Ex 4: Find the six trigonometric values (if they exist) for the point on a unit circle given below.

P 
8
15
,
17
17

3
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
Ex 5: Refer back to example 4. You were given 𝑃(𝑡) and found the six trigonometric values.
Find the following using that point 𝑃(𝑡).
(a) P(t   )
(b) P(t   )
1
1
,
2
2
 or 
2
2
,
2
2
(d ) P(t   )
  (corresponding to 30°),
 (corresponding to 45°, and  ,  (corresponding to 60°).
The most often used points in quadrant I are

(c) P(t )
3 1
,
2 2
1
2
3
2
Every angle is quadrants II, III, and IV can use an acute angle, called a reference angle to
help find the trigonometric value of θ. Let θ be a non-quadrantal angle (not 0, 90, 180, or 270
degrees) in standard position. A reference angle of θ is the acute angle θR that the terminal
side of θ makes with the x-axis. (See the figure on the next page and the definitions in the
box on the next page.)
This reference angle discussion is continued on the next page.
4
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
y
x
If θ is in Q I, θR = θ
If θ is in Q II,  R  180   or   
If θ is in Q III,  R    180 or   
If θ is in Q IV,  R  360   or 2  
Don’t try to memorize these. Use logic. Always find the difference between
the angle and the closest positive or negative x-axis.
Ex 4: Find the reference angle for each θ.
a) 355
b) 120
d ) 255
e)  100
c) 12
g ) 785
f )  215
h)  412
5
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
Let’s use our TI-30XA calculator to compare the following.
(a) cos 33 and cos(33 )
(b) sin18 and sin(18 )
(c) tan 63 and tan(63 )
You will notice that the values in each pair in part (b) and (c) were the same. The picture below
explains why. An acute angle θ and the angle – 𝜃 are in quadrant I and IV. The sine and tangent
values of θ and –θ in those two quadrants are opposites.
You will notice that the values in the pair of angles in part (a) were the same. That is because the
cosine values of angles θ and –θ in those quadrants are equal.
Note: If θ and –θ were in QII and Q III, etc., there would be the same results.
The arguments above lead to the following formulas.
Formulas for Negatives:
cos( )  cos 
sin( )   sin 
tan( )   tan 
6
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
Ex 5: Use the formulas for negatives to find the exact values of the following.
 
(a) sin    
 6
(c) cos  135  
 
(b) tan    
 2
(d )
cot  45  
Ex 6: Verify this identity by transferring the left side to the right side.
csc( x) cos( x)   cot x
To find Trigonometric Values of an Angle:
1.
Determine the quadrant where the terminal side of the angle is located.
2.
Find the reference angle.
3.
From the quadrant, determine if the trig value is positive or negative.
4.
Find the value with the correct sign.
Ex 7: Find the exact trigonometric values.
(a) sin135 
(b) cos 240 
7
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
(c) tan(210 ) 
(e) sin
5

4
 11
( g ) sin  
 6
(i ) tan
7

6
(k ) cot
5

3



Summer 2016
(d ) cos(390 ) 
( f ) tan
11

3
( h) cos
7

4
( j ) csc
3

4
 5
(l ) sec  
 3



8
MA 15800
Lesson 20 Notes
The Unit Circle & Reference Angles
Summer 2016
When using your TI-30XA calculator to approximate a trig value:
1.
Put your calculator in the correct mode (degrees or radians)
2.
If necessary, convert minutes and seconds to part of a degree.
3.
If finding the sine, the cosine, or the tangent value; press the correct key.
4.
If finding the secant, the cosecant, or the cotangent value; find the sine,
cosine, or tangent value first; then use the reciprocal key.
Ex 8: Approximate toe four decimal places.
(a) cos(15 47) 
(b) sin(2.34) 
(c) cot (127 28) 
(e)
tan(75 ) 
(d ) sec(0.45) 
( f ) csc(5.26) 
9