Trigonometrical functions come not to all school students easily. And if still it is possible to cope with the equations somehow by means of formulas, then to construct the schedule of cos or sin for some it seems an impossible task. Meanwhile, for this purpose it is only necessary to know an algorithm of creation of schedules of trigonometrical functions.
It is required to you
- - the sheet of paper (it is better in a cage);
- - ruler;
- - pencil and handle;
- - eraser;
- - calculator.
Instruction
1. Draw axes of coordinates. ou postpone for axes values +1,-1 and divisions between them (if cos is increased by a large number, for example, on 5, then mark an axis up to +5 and-5). Oh postpone values for axes x, multiple to value π (for example, postpone 2π, π, π/2, π/4, π/6).
2. Put the main end of the schedule of cos: these are points with coordinates (π/6; 0.87), (π/4; 0.7), (π/3; 0.5), (π/2; 0), (π;-1), (3/2 π; 0). For more exact schedule take the calculator and substitute any values x in the cos function. For example, to count value at in a point 0.8π gather in the calculator number 90 (value π in degrees), increase it by 0.8 and take cos. Round the received value to 0.3 and put the end (0.8π; 0.3) on the schedule. On noted points carry out a smooth curve.
3. Consider that the schedule of cos belongs to periodic therefore there is no need to build the schedule of big length. Construct a piece from 0 to 2 π and duplicate its necessary number of times.
4. If to the cos function the number is added, for example, it has an appearance at = cos x +1, then the schedule needs to be lifted up on this number. Accurately, without breaking proportions, transfer all control points to necessary value up (in other words, add to value at this number). If a negative number (at = cos x-3), then, respectively, lower the schedule.
5. To construct the function graph increased by some number, for example, at = 2 cos x, stretch the schedule on an axis ou, that is at increase all values by the required number (if it is simpler to tell, "mountains" will become higher than your schedule, and "holes" are lower). Consider if number before cos less than 1, then the schedule, on the contrary, becomes more flat.
6. The third case is a schedule with a multiplier before x, for example, at = cos of the 2nd. For creation of such schedule, stretch standard curve cos on an axis oh in the necessary number of times (in an example – twice). Consider that if number before x less than 1, then the schedule, on the contrary, contracts.
7. If to value x in cos any number, for example, at = is added or subtracted by cos (x - π/2), then transfer the schedule horizontally to this number.
8. If you were given a task to construct not just the schedule at = cos x, and more difficult option, then perform all operations by a pencil that they could be erased subsequently. How function looks, change the schedule, at the same time carry out all changes consistently. For example, if function looks at = 3*cos of the 2nd +5, then at first stretch the schedule on an axis oh twice, then stretch it on an axis ou by 3 times, and in the last turn raise it by 5 units up.
9. After all manipulations are done with the schedule, substitute some value in function and find coordinates of one point. If it coincided with your schedule, everything means it was done correctly, circle the cos line with the handle and you sterit all auxiliary lines.