Circular Functions Just a fancy name for Trigonometry

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 1

Intro: Prior Learning Is the Greek letter ‘Theta’. Tend to represent angles by Greek letters SO/H CA/H TO/A Opposite e s u n e t o p y H Adjacent 2

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: 4 820 10 4210 5 A = sin– 1(10/20) (5/10)==30° (4/8) (2/4) =30° A 3

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. 2 Given sin(A) = 0. 5 you know the Opposite side from the angle is half the length of the hypotenuse 4 A. 5 1 A Or it could be this similar triangle (the exact same shape). 4

Know the Trig ratio – Know the shape 2 5

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

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! 7

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’. A radian is about 57. 3°, or close to 60° s diu 1 ra 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. 9

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

Unit Circle and Standard Angles y r= 1 0, 1 Point on a circle – 1, 0 We always measure angles in a counterclockwise direction from the positive xaxis to the terminal arm x 0, – 1 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) 11

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

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

Co-terminal Angles 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! Coterminal Angles (Radians): 13 /4 5 /4 – 3 /4 5 /4 + 2 = 5 /4 + 8 /4 = 13 /4 5 /4 + 2 = 21 /4 5 /4 – 2 = 5 /4 – 8 /4 = – 3 /4 5 /4 – 2 = – 11 /4 Coterminal angles given by: + n*2 where n is any integer (positive or negative) 14

Reference Angles Reference Angle [ 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. Standard Angle 40° ref nearest x-axis 140° Standard Angle nearest x-axis 230° 50° ref 15

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

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

Special Angles - Quadrantal Angles 18

Special Angles 45°– 45° Triangle We know the value of the sine and cosine of a 45° – 45 ° – 90 ° triangles from Geometry! x 45° 1 1 Check with your calculator! sin 45° is approx 0. 707 and so is 19

Special Angles 30°– 60° Triangle We know the exact value of the sine and cosine of the 30°– 60 – 90° triangle from geometry 60° 2 2 30° 1 x 2 60° 20

Exact Trig Values Pattern 0 30 45 60 90 0 /6 /4 /3 /2 sin 0 1 cos 1 0 0 30 45 60 90 0 /6 /4 /3 /2 sin cos 21

Reading Sine and Cosine from Unit Circle Chart (cos 90, sin 90) (cos 60, sin 60) (cos 135, sin 135) (cos 45, sin 45) (cos 30, sin 30) 22

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

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

Graph of Cosine ‘Wavelength’ or ‘Period’ of one cycle Cosine is just the x-coordinate on the unit circle; how far left or right you are of the ‘hub’ 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: -1 y 1, you will never go above 1 or below -1 25

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

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! 29

Reciprocal Trig Function - Cosecant y 1/4 4 1/2 2 Sine curve 2 1 asymptote Plotting csc in red 4 3 y=1 y=0. 5 y=0. 25 2 Smaller fractions on the sine curve, turn into larger numbers on the cosecant curve 3 30

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