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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