Transcript Slide 1
Don’t you just love those great questions in the exam, but every time struggle to get the right answer? So how do we do this? Now first things first, where are the multiplication signs? Remember, there is no multiplication between the cosine and the bracket. Why? “Cos” I said so! Ha-ha………… No, because Cos(3x-30°) is a function like f(x) or g(x) where Cos is the name of the function and (3x-30°) is the input value of the function. Further more we need to make up for the lazy mathematicians that always left the brackets out. Let us make sure that the input values of the trigonometry function are in brackets. So now we can work with an equation The idea is to get the “unknown terms” on the left and the “known terms” on the right Subtract 3 from both sides Divide both sides by -3 Now that the “unknown terms” are on the left and the “known terms” on the right we can solve for x by: applying the input value to the rule of the function We have all the ingredients together and can start to mix it together. And like any good recipe we need to follow the steps carefully to ensure that the cake may come out correctly. To solve this we have to start with the standard form of the cosine function. Because Mathematics is such a cute and positive subject we always start with the positive acute angle of the standard function. Quadrant I Quadrant II Quadrant III Quadrant IV 70,5° This is called our reference angle Still looking at the standard form of the cosine function. We now consider how we could have reached the correct answer by using the reference angle within one period and set that equal to my original function’s input value: Period = 360 ° Quadrant I Quadrant II Quadrant III 180°-70,5° Quadrant IV 180°+70,5° X X OR The input value of the function How I used the reference angle to reach the solution The same solution will repeat itself every period K represents the periods and must be an integer. Again we have equations that we need to solve: OR Simplify the brackets: OR Add 30° to both sides: OR Divide both sides by 3: OR Simplify the fraction: OR Remember we started with the following problem: Then we collected all the ingredients: Mixing all the ingredients we landed up with: OR Now we need to bake the cake. We need to find all the possible solutions within the restrictions given. To do this we need to replace the k-value in both cases and see what values of k satisfy the restrictions. Some people prefer to bake in the normal oven and some still use the old ant hill oven on the farm. I know the purists will debate that a cake baked in the anthill oven taste so much better but I prefer the microwave oven. (My trusted Casio fx82ES Plus!) I believe that people do not use it as they do not know how to or the amount of time they can safe. So how do we do this: 1. 2. 3. 4. Remember to switch your calculator on. Press the mode button Press 3 to select the table mode Complete the f(x)= by putting in the first option a) Press 46,5 + b) then the alpha key c) and then the “x” key for the k-value d) Then press (120) e) It will read as follow when you press = button 5. When asked where to start: press the (-) button and 3 or any number you prefer, then press the = button 6. When asked where to end: press 3 or any number you prefer, then press the = button 7. When asked in what steps: press 1 or any number you prefer, then press the = button 8. Now you will have a table with possible solutions 9. Record your solutions and select the ones that fall within your restrictions. 10. Repeat the same for the second option. And now the cake can come out of the oven! Here then the table with possible solutions. Use the restrictions to determine the possible solutions. OR X= -73,5° X= 46,5° X= 166,5° X= -146,5° X= -26,5° X= 93,5° Herholdt Bezuidenhout [email protected]