Download Coordinate Plane Trig Circles

Coordinate Plane
Trig Circles
The coordinate plane has long been an important tool in mathematics. In this activity, you will use the coordinate plane to
see a connection between three topics you’ve studied in past math classes; The Pythagorean Theorem, the Distance
Formula, and the Equation of a Circle. From prior math classes you should remember these;
Distance Formula:
Pythagorean Theorem:
2
d  ( x1  x2 ) 2  ( y1  y2 ) 2
2
a +b =c
Equation of a Circle:
2
( x  h) 2  ( y  k ) 2  r 2
If you simply rewrite these (using a little algebra) a little differently, you’ll see how they are really different applications of the same
equation (especially when placed on the coordinate plane);
Distance Formula:
Pythagorean Theorem:
( x1  x2 )  ( y1  y2 )  d
2
2
2
2
Equation of a Circle:
2
( x  h) 2  ( y  k ) 2  r 2
Leg + Leg = hyp
2
Background Info – part 1
When a Line is rotated from the x-axis and it makes a circle that has a radius
of r. If a point P lies on the terminal side of the angle, then it has coordinates
(x,y) and they form the legs of the right triangle with the circle’s radius serving
as the hypotenuse;
Pythagorean Theorem:
2
2
y
P
2
Leg + Leg = Hyp
y
r
Equation of the circle centered at (0,0):
2
X
+ y2 = r2
2
X
2
2
+y =d
Background Info – part 2
Using an ordered pair, consider a point P on a coordinate plane so that it has
an ordered pair address (x,y). Extend a line from the origin of the coordinate
plane through point P. With this line constructed, the terminal side of an angle
() is formed. IF a circle were to pass through this point then the distance from
the origin to the circle would be a radius. If you consider the x & y coordinates
of the point as “legs” of a right triangle, then you have what is sometimes
called reference triangle. You were introduced to the ratios of triangles in
geometry called the Sine, Cosine, and Tangent. They are shown here for
your reference, along with three new “reciprocal” ratios.
y ,
x ,
y
sin  
cos 
tan  
r
r
x
There are three reciprocal ratios;
r
r ,
sec  , cot  x
csc 
x
y
y
x
x
It is also the distance between the point P and the origin
y
P (x , y)
r
y

x
x
Assignment – part 1
For each point from the coordinate plane listed, find the radius of the circle that would pass through it. If a
radius is given find the missing coordinates of the point that would lie on that circle.
1. P( 3, 4)
5. P( .5 , .866)
2. P( -5 , 12)
6. P( x, 2) r  13
3. P( 2, -3)
7. P( 5, y) r  89
4. P( 1, -1)
8. P( a, b)
Assignment – part 2
For the first set of problems where you found the radius of the circle that would pass through a point on the coordinate
plane, find the six trigonometric ratios for each point.
9. P( 3, 4)
10. P( -5 , 12)
11. P( 2, -3)
12. P( 1, -1)
13. P( .5 , .866)
14. Compare the values of the trig functions “sine”, “cosine” and “tangent” for points A, B, & C by completing a table
showing the point, it’s sine value, cosine value and tangent value. Then compare the trig functions of points M, N, & Q.
Point
A (3,4)
B (-3,4)
C (-3, -4)
D (3, -4)
Sin 
Cos
Sin 
Point
tan
Cos
tan
M (12,5)
N(24,10)
Q(- 48, 20)
What do you notice about their respective values? Are there any patterns, similarities seen? How are they
different? Do they have any common or similar traits?
15. Since points on the coordinate plane carry positive and negative values, use any points from the coordinate plane
you would like to determine what happens to the trig function in each quadrant. Just keep in mind that all you
should be concerned about is the “positive” or “negative” sign of the numbers you use. Put your answers in a
completed table like the one shown:
Trig Functions & their signs
Quadrant I
x y r
+ + +
Sin = y/r
r
Csc = / y
Cos = x/r
r
Sec = /x
Tan = y/x
x
Cot = / y
Pos +
Quadrant II
Quadrant
III
Quadrant
IV
x
x
x y r
+ - +
-
y
+
r
+
-
y r
- +
Find the exact values of the five remaining trig functions without finding  given the information about one
of the trig functions and secondary information about another trig function for the same angle  Use the
Pythagorean theorem and your knowledge of positive and negative values to solve these problems.
(hint: 1 is always a denominator of any number when nothing is written & “>0” means “positive”)
16. cos   3 ;  is a quadrant IV angle
17. sin   3 ;  is a quadrant II angle
18. cos   3 ;  is a quadrant IV angle
19. csc    5 ;  is a quadrant III angle
20. sin    2 ;
21. cos    3 ;
5
5
3
22. sec 
3;
cot   0
sin   0
5
4
5
23. tan   2;
tan   0
csc   0