Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics The Unit Circle A learning resource prepared for the Mathematics Help Centre of the School of Mathematics, University of South Australia Contact: Garry Lockwood, University of South Australia

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Aim Promote an understanding of the unit circle and hence the trigonometric functions it is used to define Use this understanding to solve problems that require a knowledge of the unit circle.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Why study the unit circle? Without understanding the unit circle one cannot understand the trigonometric functions sin, cos and tan, which are essential to solving many problems in trigonometry. Examples include: Solve the triangle with sides a=9.4, b=13.1 and angle A=36  Finding a Fourier series to approximate a given function Adding together two sinusoidal waves, say Asin(bt)+Csin(dt) to get a single wave of the form Esin(ft+g)

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics What is the unit circle? Simply, the unit circle is a circle centered on the origin, (0,0), with radius 1. The equation of a circle with centre (h,k) and radius r is (x-h) 2 +(y-k) 2 =r 2. Hence the equation of the unit circle is x 2 +y 2 =1. Click here to run the program DrawCircles.exe which will plot the graphs of circles of various centres and radii. (Note that the graphs may not not look perfectly round due to problems with the scale.)here

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics The unit circle and the point P We are going to use the unit circle to define the trigonometric functions in quite a sneaky way. Let P be a point on the circle itself and draw a line from the origin to the point P Let  be the angle between the positive x-axis and the line joining P to the origin, as shown in the next slide.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Joining P to the x and y axes We started out by defining a point P on the unit circle and named the angle between the line segment joining P to the origin. Drop horizontal and vertical perpendiculars from P until they meet the x and y axes, as shown in the following diagram.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics P joined to the axes

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Defining the trig functions Let x and y be the horizontal and vertical coordinates of the point P, or equivalently let x and y be the points on the horizontal and vertical axes where the perpendiculars cross We then define the trig functions as follows: sin(  )=y cos(  )=x tan(  )=y/x

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics So What? Armed with these definitions we are now able to calculate some basic trigonometric values. You should view the next presentation, Basic_Trig_Values for details of how we calculate some values and how we use the unit circle concept to extend these values to cover a much larger domain.