Objectives New learning Understand use the sine cosine

Objectives New learning Understand use the sine, cosine and tangent functions; their graphs, symmetries and periodicity. Take into to a new level – understanding them in terms of radians Other outcomes Revise year 12 work COPY TITLE: sine, cosine and tangent functions and graphs

Year 12 Review

Become one with the sine and cosine function ….

Remembering how to label the triangle

Remembering the basics …. Going backwards Remember you must be working in degrees

Remembering the basics cos Going backwards Remember you must be working in degrees

The sine function – an informal approach (Getting ready for the unit circle …)

The sign curve is a graphical representation of what is happening to sine as it goes through the angles ….

Graph of y= sin x Complete the projection of sin values from the unit circle onto the Cartesian plane on the right and then join the points with a smooth curve. 09: 38 Lesson interaction Introduction

Cosine curve using the unit circle method

Sine Curve Cosine Curve Sine and Cosine Curve

Why do we need to know these curves (in the context of C 2) If you put sin 40 into your calculator you get 0. 642787809…… In reality if you look at a sine curve there is more than one occasion when the sine curve is equal to sin 40 (ie 0. 642787809…. . ) and you need to be able to identify the other angles. (The calculator wont give you them but you can use it to check) Once you have worked one out you can then use you calculator to check

Use your graph to find al the angles equal to sin 50

Year 13 Now you have to be able to understand the curve in terms of radians

Now we are just going to remind ourselves of solving trig equations (think about the development of using radians)

(We are working in degrees for the moment – until we have reviewed this work)

Thinking about stretch factors …. . When you have a function f(x) and it is transformed to f(2 x). This is stretch parallel to the ___ axis It is a stretch factor of _____ The effect of this on the curve is _______ Let’s have a look at this magic …….

Graphs of Related Functions (7) y. The = f(x) rest of this lesson Stretches in x 2 is about how to overcome these difficulties f(x) = Sinx f(x) = Sin 2 x 1 -360 -270 -180 -90 0 -1 Let’s think about the problem this leaves us with when we are trying to find all angles? 90 180 270 x 360 Two main issues – all the values change -2 All x co-ordinates x 1/2 - The number of possible solutions changes

Let’s deal with one problem at a time …. First we will look at how the values are affected

Idea 1 Solving a tricky trig equation …. Question – Solve the equation sin 3 x = 0. 71 Step 1: sin 3 x = 0. 71 Step 2: 3 x =sin -1 (0. 71) Step 3: Find the other key angle for 3 x =sin -1 (0. 71) by looking at sin x Step 4: Get lots of other angles(+360 or -360) 3 x= 45. 2349 Formula for working out multiple of sin Family 1: sin (360 n+x) Family 1: sin (180+360 n-x) Or 3 x = 134. 8 3 x= 45. 2, 405. 2, 765. 2, 1125. 2, 1485. 2, 1845. 2 Or 3 x=134. 8, 494. 8, 854. 8, 1214. 8, 1574. 8, 1934. 8 Step 5: You have worked out 3 x – you need to find x Let’s now look at the range issue …. .

But of course the issue is making sure you catch all the solutions in a given range …. .

Remember the cos curve is different. It is symmetrical about the y axis It is an even function (You can use the same formula)

Step 1: cos 2 x = 0. 5 Step 2: Solve basic 2 x =cos -1 (0. 5) Step 3: Find other important cos value. (use symmetry of curve) Step 4: Change the range so it works for transformed curve Step 5: Find all possible values within range Step 6: Change all your values so you have x not 2 x

Now we are going to do one that looks more difficult – but it is just as easy … If you can we will see if you can try it on your own

Step 1: rearrange the equation Step 2: Solve 4 x = ______ -49 Step 3: Find the other critical value 229 Step 4: Sort out range Step 5: Find both families of solutions within range Family-49: 4 x = -409, -49, 311, 671 Family 229: 4 x = -491, -131, 229, 589, Family-49: x = -102, -12, 78, 168 Family 229: x = -123, -33, 57, 147

Now lets work on some questions from the book Page 291 Exercise 6 A

You use the same basic strategy when dealing with addition (or subtraction)

Step 1: sin (x - 60) = 0. 4 Step 2: (x - 60) =sin -1 (0. 4) Step 3: Find other critical ang Step 4: All angle in range Step 5: Adjust so that you have x not x - 60

1. Rearrange to get in form cos (90+x) 2. Sort out the range 3. Find value first value of 90 + x 4. Find other critical val. 90 + x 5. Find all values of 90 + x 6. Adjust so you have just x Answers : 36. 9 , 143. 1