Trig. Trig is also covered in Appendix C of the text. 1SOHCAHTOA. These relations were first introduced

Transcription

1 Trig Trig is also covered in Appendix C of the text. 1SOHCAHTOA These relations were first introduced for a right angled triangle to relate the angle,its opposite and adjacent sides and the hypotenuse.

2 sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj Example: In a right-angled triangle with a 30 angle opposite a 4 metre side, what are the other sides and angles? The next concept to tackle is the angle measurements in degrees versus radians. Unlike degrees, radian an- 2

3 gle measures are a natural part of the mathematics. 2 What s a radian? Draw a circle with an x-y axes in the centre Cut a string the same length as the diameter. Cutthatstringinhalfsonow you have a string that is one radius long. If you lay your one radius long string down on the 3

4 circle, the angle subtended by that string is a one radian angle. This translates to approximately degrees regardless of the size of the circle you drew. Neat eh? How many radius string lengths it takes to go completely around the circle? If you measured perfectly youwouldget2π times. That is, a full circle is 2π radians around. We also know a full circle 4

5 is 360 degrees around so 360 =2π radians You can use this equation to convert to and from degrees to radians. We will always talk about angles as measured in radians but you need to convert to degrees to converse with the outside world. You can t tell the cab driver make a π 2 radian turn at the next light. Now trig is always first introduced to you in high school 5

6 as a triangle thing. (You remember soh cah toa?) The truth is, trigonometry is very connected to the circle, a shape ancient people considered perfect because it had no corners or edges, no beginning or end. Say our circle has a radius of 1 unit. We can then call it a unit circle. Any point on that circle has x and y coordinates. The x coordinate of any point on the unit circle is cos θ, the y coordinate is sin θ. 6

7 That s where the cos θ and sin θ functions come from. Cast Rule? We will now see that everything falls out of that unit circle. Takea90degree(or π 2 ) angle for example. The coordinates on the unit circle are (0, 1), therefore the 7

8 cos(90 )=0, sin(90 )=1. Try it on your calculator. What about the angle 270 = 3π 2? Consider the angle 90 There are six trig values, cos(θ), sin (θ), tan(θ), sec(θ), csc(θ) and cot(θ). They are related by the 8

9 equations: sin (θ) tan(θ) = cos(θ) sec(θ) = 1 cos(θ) csc(θ) = 1 sin(θ) cot(θ) = cos(θ) sin(θ) = 1 tan(θ) So, once we know the cos(θ) and sin(θ) from the x and y coordinates, we can calculate theother4trigvalues. Example: What is the cot( 270 )? Also, do you remember the equation for a circle with 9

10 centre (0, 0) andwithradiusr? Yes, x 2 + y 2 = r 2. Well, in our case the unit circle has radius 1 so the equation of the unit circle is x 2 +y 2 =1. Noting that x =cos(θ) and y =sin(θ) we can see that cos 2 (θ)+sin 2 (θ) =1 That is the father trigonometric identity. Divide that whole identity equation by cos 2 (θ) to get cos 2 (θ) cos 2 (θ) + sin2 (θ) cos 2 (θ) = 1 cos 2 (θ) 1+tan 2 (θ) =sec 2 (θ) 10

11 or divide that whole identity equation by sin 2 (θ) and get cos 2 (θ) sin 2 (θ) + sin2 (θ) sin 2 (θ) = 1 sin 2 (θ) cot 2 (θ)+1 = csc 2 (θ) This is how we derive the three identities which you should memorize. cos 2 (θ)+sin 2 (θ) =1 1+tan 2 (θ) =sec 2 (θ) cot 2 (θ)+1 = csc 2 (θ) See how cos(θ) is related to sin(θ), tan(θ) to sec(θ) and cot(θ) to csc(θ). This is a pattern that will reoccur when we learn the 11

12 derivatives of trig functions. But wait, you say. I can only get the x and y coordinates for 4 points, where the unit circle crosses the axes. What about other angles like 45? First memorize these two right angled triangles Notice their hypoteni are both 1 unit long. Redraw them 12

13 with the radian angle measures so we can use them in the next few examples. Example Q? Find cos( π 6 ) exactly. A. We need to find the x-coordinate that corresponds to the angle of π 6 =30. Using the second triangle above, we can fit it into the unit circle. Now you see why the hypotenuse is one unit, so fits exactly into our unit circle. 13

14 We can see that the x- coordinate is 3 2. Therefore cos( π 6 )= 3 2. You can confirm the answer to the above Example on a calculator by getting a decimal for 3 2 and a decimal for cos( π 6 ). Remember to have your cal- 14

15 culator in radian mode. If the two decimal approximations are the same, then you can be happy with your exact result. Example Q? Evaluate sec(225 ) exactly. 15

16 3 Trig Graphs The unit circle also helps us understand why the trig functions have the graphs they do. The x and y coordinates both oscillate between +1 and 1 as the angle increases from zero. So do the graphs. y =cos(x) x y =sin(x) 16

17 x Show using the unit circle that cos(x) is even and sin(x) is odd. Notice these facts correspond to the graphs. Wewilllearntograph more trig functions later The other memorization I want you to do is these double angle formulae. 17

18 sin(2x) =2sin(x)cos(x) cos(2x) =cos 2 (x) sin 2 (x) =2cos 2 (x) 1 =1 2sin 2 (x) These double angle formulae will help you later in higher order calculus courses. And the answers in the back of the book are at times the result of a major simplification using these trig equations. Now some exercises to solidify your new trig knowledge. 18

19 Do exercises.1 to 18,28to32inAppendixC of the text. Submit numbers 4,10,12,14,18,28,30,32 neatly at the beginning of class on Friday. 19