Trigonometry in the Coordinate Plane PART ONE Trigonometry

Trigonometry in the Coordinate Plane PART ONE

Trigonometry in Coordinate Plane First a review of what we know. In any right triangle we can label the sides opposite, adjacent and hypotenuse based on a given angle θ. The trigonometric ratios are then defined as hyp opp Sin θ = opp/hyp Cos θ = adj/hyp Tan θ = opp/adj θ adj

Trigonometry in Coordinate Plane Now we are going to create a triangle in the coordinate system and see how we can write new definitions for sine, cosine and tangent. We will start by making an angle θ in the coordinate system by drawing a line segment starting at the origin as seen to the right. The end of the line segment is at the point (x. y) also shown in the diagram.

Trigonometry in Coordinate Plane To make the right triangle we draw a line segment down from the point (x, y) to the xaxis We label the sides based on the angle θ as before - opposite, adjacent and hypotenuse And we can write the same trig functions as before Sin θ = opp/hyp Cos θ = adj/hyp Tan θ = opp/adj

Trigonometry in Coordinate Plane Now, lets suppose that the point (x. y) was (5, 7) That would mean you would have to move 5 units in the x direction and 7 units in the y direction to get from the origin to the point (5, 7) Looking at our triangle we can see a side that moves in the x direction (adj) and a side that moves in the y direction (opp) So that means that the adjacent is 5 and the opposite is 7 as labelled in the diagram.

Trigonometry in Coordinate Plane Now we could replace (x, y) with any values we choose but to keep it general we are going to stick with (x, y). From the last slide we know that we can call the adjacent side x and the opposite side y which we have done in the diagram to the right

Trigonometry in Coordinate Plane One last change! We can imagine drawing a circle with its center at the origin and going through the point (x, y) This would make the hypotenuse the radius of that circle so we can change its label to r for radius.

Trigonometry in Coordinate Plane

Trigonometry in Coordinate Plane Lets stop for a minute and be sure we understand that nothing has been changed except for labels. On the left is the original picture we had in the first slide and on the right is the picture from the last slide. All we have changed is the names of the sides from opposite, adjacent and hypotenuse to y, x and r. hyp θ opp adj

STOP IF YOU ARE NOT SURE OF WHAT JUST HAPPENED GO BACK AND READ THE PREVIOUS SLIDES AGAIN. ALTHOUGH WE HAVE ONLY CHANGED THE LABELS, BY DOING SO WE ARE NOW GOING TO CREATE RELATIONSHIPS THAT WERE NOT VISIBLE PRIOR TO THIS CHANGE. ONCE YOU ARE READY LETS MOVE ON.