1 Circular Functions Just a fancy name for Trigonometry –Thinking of triangles as central angles in a circle Have done trigonometry ever since grade 9 This slide show assumes a moderate recollection of previous grades Trig It assumes you have completed Unit B Transforms –Although you will survive without Unit B Click the small icon of the projection screen to animate this slide show. Enjoy

2 Intro: Prior LearningSO/H CA/H TO/A   Is the Greek letter ‘Theta’. Tend to represent angles by Greek letters Hypotenuse Opposite Adjacent

3 Intro: Prior Learning 2 Hopefully you are familiar with trigonometric values in a table form: A triangle with a corner angle whose ‘sine ratio’ is 0.5 for example has an opposite side that is half the length (0.5) of the hypotenuse. Since ‘sine’ is just a ratio of the length of the opposite side divided by the length of hypotenuse side of a right angle triangle. In other words, sine is how many hypotenuses fit into the opposite side. Here are triangles with a corner angle, A, that has a sine of 0.5:105 A = sin –1 (5/10) = 30° A = sin –1 (4/8) = 30°A = sin –1 (2/4) = 30°A = sin –1 (10/20) = 30° A

4 Know the Trig ratio – Know the shape 1 It is easy to calculate a trig ratio; just the length one side of a right angle triangle divided by the length of another side. So if you are given the lengths of some sides it is easy to calculate. But the reverse is true also, if you know a trig ratio of the right angle triangle, you know at least the relative shape of the triangle. Given sin(A) = 0.5 you know the Opposite side from the angle is half the length of the hypotenuse A 4 2 A 1.5 Or it could be this similar triangle (the exact same shape).

5 Know the Trig ratio – Know the shape 2

6 Why do we spend so many grades studying triangles again?

7 Measuring angles You are familiar with how to measure angles using a protractor and how to find the measure of an angle when given its trig ratio using the inverse functions: cos -1, sin -1, tan -1 on your calculator (or looking up the angle backwards in a table). Example: on your calculator cos -1 (0.5) = 60° which means the angle that has a cosine of 0.5 is 60 °. But just like you can measure distances in units of metres, or feet, or miles there is another unit by which to measure angles, and it is not degrees!

8 Only Obtuse angles in a right angle triangle

9 Radians – The universal measure of angles Only an earthling would use 360 degrees as the measure of the full angle around a circle. Only earth goes around the sun every 360 days (ish), only earthlings count with 10 fingers and use decimals, etc. The proper mathematical and ‘non- prejudiced’ way to measure angles is to use a measure of a central angle called a ‘radian’. 1 radius An arc on the circumference of the circle of length 1 radius   = 1 radian or 1 r or just ‘1’ If you move around a circle a length of 1 circle radius, you have moved through an angle of 1 ‘radian’ about the centre. 1 radius A radian is about 57.3°, or close to 60°

10 Radian ConversionHow to convert between Degrees and Exact Exact Radians Radians: There are 100 cm to 1 meter There are 12 things to a dozen There are 2 2 radians to 360° of a circle 180° is the same as rr Conversions: Radians = Degrees * r r / 180° Degrees = Radians * 180 °/ rr Converting an angle to exact radians means the angle will have in it

11 Unit Circle and Standard Angles x y 0,1 –1, 0 0, –1–1 1,0 Point on a circle r =1 We always measure angles in a counter- clockwise direction from the positive x- axis to the terminal arm  It is often nice to work with a ‘Unit Circle’ with a radius of 1. It just makes calculations easier. Superimposed on a grid the unit circle passes through the points : (1, 0); (0, 1); (-1,0); (0,-1)

12 Unit Circle Chart See how the common radian measures are converted to degrees on this chart. 60° is  /3 The points in brackets are discussed later

13 Co-terminal Angles Coterminal Angles Coterminal Angles. Standard angles that share the same terminal arm  225° 585° Coterminal Angles (Degrees): 225° 225° + 360° = 585° 225° + 360° ° = 945° 225° 225° – 360° = –135° – 360° – 360° = –495° Coterminal angles given by: + n*360° n*360° where n is any integer (positive or negative) –135°

14 Co-terminal Angles 13  /4 –3  /4 Coterminal Angles (Radians): 5  /4 5  /4 5  /4 + 2 = + 8  /4 8  /4 = 13  /4 5  /4 5  /4 + 2 + 2 = 21  /4 5  /4 5  /4 –2 = – 8  /4 8  /4 = –3  /4 5  /4 5  /4 –2  –2 = –11  /4 Coterminal angles given by: + n*2  n*2  where n is any integer (positive or negative) 5  /4 Every angle in standard position has an infinite number of co-terminal angles; just depends on how many more times you want to wind or un-wind it!

15 Reference AnglesReference Angle [  Ref ] [  Ref ] : The positive acute (ie: less than 90  ) angle formed by a terminal arm and the nearest (positive or negative) x-axis. The angle of a simple triangle that the classical Greeks would have discussed.  nearest x-axis ref 140° 40° Standard Angle  nearest x-axis  ref 230° 50° Standard Angle

16Sine Hypotenuse Opposite Adjacentx y  0,1 –1, 0 0, –1 1,0 Classical Greek Sine Definition: Modern Circular Definition: y r The classical and modern definitions of sine agree; except we now call the hypotenuse a radius, r and we call the opposite side the ‘y-coordinate’ of a point on the circle Point on a circle P(x,y) If we pretend the circle has a radius of one then the sine is just the percentage of how high above or below below the ‘hub’ (ie: centre or origin) of the circle we are.

17Cosine Hypotenuse Opposite Adjacent x y  0,1 –1,0 0,–1 1,0 Classical Greek Cosine Definition: Modern Circular Definition: x r The classical and modern definitions of cosine agree; except we call the hypotenuse a radius, r, now and we call the adjacent side the ‘x-coordinate’ of a point on the circle Point on a circle (x,y) If we pretend the circle has a radius of one then the cosine is just the percentage of how right or left left of the centre of the circle we are. The point P(x, y) is therefore (cos , (cos , sin  )  ) on the unit circle cos, sin 

18 Special Angles - Quadrantal Angles

19 Special Angles 45°– 45° Triangle We know the value of the sine and cosine of a 45° – 45 ° – 90 ° triangle angles from Geometry! 45° 1 1 x Check with your calculator! sin45° is approx and so is

20 Special Angles 30°– 60° Triangle We know the exact value of the sine and cosine of the 30°– 60 – 90° triangle from geometry x 60° 30° 30°

21 Exact Trig Values Pattern   0  /6  /4  /3  /2 sin  01 cos  10   0  /6  /4  /3  /2 sin  cos 

22 Reading Sine and Cosine from Unit Circle Chart (cos30, sin30) (cos45, sin45) (cos60, sin60) (cos135, sin135) (cos90, sin90)

23 The Unit Circle Exact Values Later you will learn to find exact values of any angle. Not just these special ones.

24 Graph of Sin  (converting angular motion to grid coordinates) y  0 2222 Angle  [radians] 1 sin is just how high above the hub of a ferris wheel you are on a unit circle as function of your angle Sin(  ) ‘Wavelength’ or ‘Period’ of one cycle Domain Domain: All  ; ie: - - - - < <  ;  ; you can go around front wards or backwards as many times as you want Range: Range: y 1, you will never go above 1 or below

25 Graph of Cosine ‘Wavelength’ or ‘Period’ of one cycle Angle  [radians] The period or wavelength is 2  radians Domain Domain: All  ; ie: - - - - < <  ;  ; you can go around front wards or backwards as many times as you want Range: Range: y 1, you will never go above 1 or below Cosine is just the x-coordinate on the unit circle; how far left or right you are of the ‘hub’

26 Graph of Sine and Cosine Recall your Transformations Unit? Notice here that sin(  ) = cos(  –  /2) but we will save that idea for the Trigonometry Identities studies. Period or wavelength of a cycle = 2 

27 Tangent of an angle in the classical sense is defined as: In the Circular Functions sense then it is defined as : So it is really:

28 Graph of Tangent  2222 Asymptote  =  /2 Asymptote  = 3  /2 Wavelength or period  or 180°

29 Reciprocal Trigonometric Functions Remember the Transformations unit? ‘y’s less than one got stretched to big ‘y’s, (Eg: ½ becomes 2) ‘y’s more than 1 got compressed to small ‘y’s. (Eg: 4 becomes ¼) The reciprocal trig functions are the same idea!

30 Reciprocal Trig Function - Cosecant 2 2 2 2  3333 y Sine curve y=1 y=0.5 y=0.25 Smaller fractions on the sine curve, turn into larger numbers on the cosecant curve 1/2  2 1/44 Plotting csc in red a s y m p t o t e

31 Solving Trig Equations To be done in the ‘Trig Identity’ unit You will then learn to solve equations like: 4sin(2  +  /4) = 2 You might have already detected the answer(s)!  = -  /24, 23  /24, etc

32

33

34

35

36

37

38