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: Equation of a Circle:

2

21

2

21 )()( yyxxd a2 + b

2 = c

2 222 )()( rkyhx

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: Equation of a Circle:

22

21

2

21 )()( dyyxx Leg2 + Leg

2 = hyp

2 222 )()( rkyhx

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:

Leg2 + Leg

2 = Hyp

2

Equation of the circle centered at (0,0):

X2 + y2

= r2

It is also the distance between the point P and the origin

X2 + y2

= d2

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, ). Extend a line from the origin of the coordinate yplane 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 . They are shown here for Sine, Cosine, and Tangentyour reference, along with three new “reciprocal” ratios.

r

ysin ,

r

xcos ,

x

ytan

There are three reciprocal ratios;

y

rcsc ,

x

rsec ,

y

xcot

r

P

y

x

y

x

r

P (x , y)

y

x

y

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) 2. P( -5 , 12) 3. P( 2, -3) 4. P( 1, -1)

5. P( .5 , .866) 6. P( x, 2) 13r 7. P( 5, y) 89r 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 Sin Cos tan Point Sin Cos tan

A (3,4) M (12,5)

B (-3,4) N(24,10)

C (-3, -4) Q(- 48, 20)

D (3, -4)

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 Quadrant II Quadrant

III

Quadrant

IV

x y r

+ + +

x y r

- + +

x y r

- - +

x y r

+ - +

Sin = y/r

Csc = r/y

Cos = x/r

Sec = r/x

Tan = y/x

Cot = x/y

Pos +

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. ;5

3cos is a quadrant IV angle 17. ;

5

3sin is a quadrant II angle

18. ;5

3cos is a quadrant IV angle 19. ;

4

5csc is a quadrant III angle

20. 0cot;3

2sin

21. 0tan;

5

3cos

22. 0sin;3sec 23. 0csc;2tan