Download Measuring Angles

Defining the Sine and Cosine Function
The Sine and Cosine Function:
Suppose that the coordinates of a point Pt  on the unit circle are x(t ), y(t ) . Then the
sine of t (written sin t ) and the cosine of t (written cos t ) are defined by
sin t  y (t ) and cos t  x(t )
Place a generic point on the unit circle and label it (cos t , sin t ) .
Ask student for the values of sin t and cos t for several obvious points. Make sure to
emphasize the angle that t is representing.
Examples
Ask students what each of the following values is
 sin( 0)
 
 sin  
2
 sin 
 3 
 sin  
 2 
 sin( 2 )





cos(0)
 
cos 
2
cos 
 3 
cos 
 2 
cos( 2 )
Bounds on the Sine and Cosine
Ask the students what are possible values are for sin t . Is there a maximum value? Is
there a minimum value?
Ask the students what are possible values are for cos t . Is there a maximum value? Is
there a minimum value?
For all real numbers t,  1  sin t  1 and  1  cos t  1.
Ask students about the values of sin t in each quadrant.
Ask students about the values of cos t in each quadrant.
Pythagorean Identity
Ask students what they know about a point on the unit circle. Place a general point on
the circle and draw a right triangle with legs x and y. Ask the students what they know
about a right triangle.
Now point out that we know that the x-coordinate is cos t and the y-coordinate is sin t .
What can we then say about a relationship between sin t and cos t
For all real numbers t, sin t   cos t   1
2
2
Examples:
Even or Odd
Ask the students if they think that the function cos t is even, odd, or neither.
Show the student a point on the unit circle. Show this point reflected over the x-axis and
ask about the x-coordinates and y-coordinates of the original point and the reflected point.
Ask the students if they think that the function sin t is even, odd, or neither
For all real numbers t, cos( t )  cos(t ) (even) and sin( t )   sin( t ) (odd).
Examples:



Determine whether the function f ( x)  (cos x) 2 is even, odd, or neither.
Determine whether the function f ( x)  x 3 sin x is even, odd, or neither.
Determine whether the function f ( x)  x sin x is even, odd, or neither.

Determine whether the function f ( x)  cos(sin x) is even, odd, or neither.
Periodic Functions
Ask the students what a periodic function is. Remind the that we talked about this with
the Ferris Wheel problem.
A nonconstant function f is said to be periodic if a positive number T exists with
f (t  T )  f (t ) for all t in the domain of f. The smallest positive number T for which this
equation holds is the period of f.
Ask the students is they can draw the graphs of a constant function. Give the students a
few minutes to work in groups to create periodic graphs and tell the periods.
Period of Sine and Cosine
Ask the students if they think that the sine function is periodic. Ask why they believe this
to be true.
Ask the students what they think the period of the sine function is. Ask the students to
explain why this is the period.
Ask the students if they think that he cosine function is periodic. Ask why they believe
this to be true.
Ask the students what they think the period of the cosine function is. Ask the students to
explain why this is the period.
The sine and cosine function are periodic with period 2 . So for every real number t
sin( t  2 )  sin( t ) and cos(t  2 )  cos(t )
Examples:
Finding Other Values of Sine and Cosine
Tell the students that there are some values of sine and cosine that are easy to find.
We need to be able to find values of sine and cosine for other angles. For this we are
going to have to remember some topics from geometry.
Ask the students what they remember about isosceles triangles. Emphasize that there are
at least two equal sides, the angles opposite the equal sides are equal and the angle
bisector of the angle opposite the non-equal sides will bisect the opposite side.
Draw a picture to help them picture this information about isosceles angles.
Ask the students to find the values for sine and cosine of

.
4
Have a student explain what picture and process was used to find the values.
1
1
 
 
sin   
and cos  
2
2
4
4
Ask students to find the values for sine and cosine of

.
3
Have a student explain what picture and process was used to find the values.
3
  1
 
and cos  
sin   
2
3 2
3
Ask students to find the values for sine and cosine of

.
6
Have a student explain what picture and process was used to find the values.
3
  1
 
sin    and cos  
2
6 2
6
Examples:
Reference Numbers
Ask students where
3
3
is. Ask the students what sin
is. Ask how this was
4
4
determined.
Reference Number
For any real number t, the reference number r associated with t is the shortest distance
 
along the unit circle from t to the x-axis. For any t, the reverence number r is in 0,  .
 2
Explain that we will be able to use the reference number and the quadrant in which the
angle falls to determine sine and cosine values.
Examples:


Find the reference number r for the value t 

Find the reference number r for the value t 

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t  

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t
4

3
2

3
3

4
7

6
4

3
7

4
11

6

3
5

6
4

3
5

4
11

6
19

3


11
3
23
Find the reference number r for the value t  
4
Find the reference number r for the value t  
General Examples:
Reference Number Problems:


Find the reference number r for the value t 

Find the reference number r for the value t 

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t  

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t

Find the reference number r for the value t
4

3
2

3
3

4
7

6
4

3
7

4
11

6

3
5

6
4

3
5

4
11

6
19

3
11

3
23
4
 45 
 150 
 240 
 330 
 60 
 135
 210 
 315

Find the reference number r for the value t  

Find the reference number r for the value
Find the reference number r for the value
Find the reference number r for the value
Find the reference number r for the value
Find the reference number r for the value
Find the reference number r for the value
Find the reference number r for the value
Find the reference number r for the value







Sine and Cosine Value Problems:












 
 
Find sin   and cos 
6
6
 
 
Find sin   and cos 
4
4
 3 
 3 
Find sin   and cos

 4 
 4 
 2 
 2 
Find sin 
 and cos

 3 
 3 
 4 
 4 
Find sin 
 and cos

 3 
 3 
 7 
 7 
Find sin 
 and cos

 6 
 6 
 11 
 11 
Find sin 

 and cos
 6 
 6 
 7 
 7 
Find sin 
 and cos

 4 
 4 
 
 
Find sin    and cos  
 3
 3
 
 
Find sin    and cos  
 6
 6
 5 
 5 
Find sin  
 and cos 

 6 
 6 
 3 
 3 
Find sin  
 and cos 

 4 
 4 
t
t
t
t
t
t
t
t

 5 
 5 
Find sin  
 and cos 

 4 
 4 
 5 
 5 
Find sin  
 and cos 

 3 
 3 
 5 
 5 
Find sin  
 and cos 

 2 
 2 
 7 
 7 
Find sin  
 and cos 

 2 
 2 
Find sin  7  and cos 7 
Find sin 8  and cos8 









Find
Find
Find
Find
Find
Find
Find
Find
Find










 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Find sin  150  and cos 150 
Find sin  240  and cos 240 
Find sin  300  and cos 300 
Find sin 540  and cos540 
Find sin  720  and cos 720 
sin 45  and cos 45
sin 60  and cos 60
sin 150 and cos 150 
sin 135 and cos 135
sin 240  and cos 240
sin 210  and cos 210
sin 330  and cos 330
sin 315  and cos 315
sin  45 and cos  45










Solving sine and cosine equations
 Find all the values of t in the interval 0, 2  that satisfy the equation
2
.
2
Find all the values of t in the interval 0, 2  that satisfy the equation
cos t 

3
.
2
Find all the values of t in the interval 0, 2  that satisfy the equation
1
sin t   .
2
sin t 


Find all the values of t in the interval 0, 2  that satisfy the equation
3
.
2
Find all the values of t in the interval
cos t  1.
Find all the values of t in the interval
sin t  1 .
Find all the values of t in the interval
t 1
cos  .
2 2
Find all the values of t in the interval
cos t  





0, 2  that satisfy the equation
0, 2  that satisfy the equation
0, 2  that satisfy the equation
sin 3t  
2
.
2
If sin t  
2 2
1
and cos t  , find the sine and cosine of the given values
3
3
a. t  

0, 2  that satisfy the equation
b.  t
c. t 

2
d.  t 

2
Find all the values of t in the interval 0, 2  that satisfy the equation
cos t 2  cos t  2  0 .

Find all the values of t in the interval 0, 2  that satisfy the equation
2sin t   sin t  1  0 .
Find all the values of t in the interval 0, 2  that satisfy the equation
(sin t )(cos t )  sin t  cos t  1  0 .
Find all the values of t in the interval 0, 2  that satisfy the equation
sin t  cos t  1.
Find all the values of t in the interval 0, 2  that satisfy the equation
2



cos t 2  cos t  2  0 .