How To Do Trigonometric Functions How To Do Trigonometric Functions
The trigonometric functions are very important in technical subjects like science, engineering, architecture, and even medicine. You will come across trig functions all the time and you are encouraged to learn them well! Surveying is one of the many applications. Road makers, bridge builders and those whose job it is to get buildings in the right place all use trigonometry in their daily work. Let's start with the basic sine function, f(t) = sin(t). This function has an amplitude of 1 because the graph goes one unit up and one unit down from the midline of the graph. This function has a period of 2π because the sine wave repeats every 2π units. The graph looks like this: Do you see that the graph is three times as tall? The amplitude has changed from 1 to 3. This is always true: Whatever number A is multiplied on the trig function gives you the amplitude; in this case, that number was 3. So 0.5cos(t) would have an amplitude of 1/2, and –2cos(t) would have an amplitude of 2 and would also be flipped upside down. Know More About Associative Property Of Multiplication Worksheets
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Do you see that the graph is squished in from the sides? Do you see that the sine wave is cycling twice as fast, so its period is only half as long? This is always true: Whatever value B is multiplied on the variable, you use this value to find the period ω (omega) of the trig function, according to the formula: In the sine above, that value was 2. (Sometimes the value of B inside the function will be negative, which is why there are absolute-value bars on the denominator.) The formula says that cos(3t) will have a period of (2π)/3 = (2/3)π, and tan(t/2) will have a period of (2π)/(1/2) = 4π. (Note: Different books use different letters to stand for the period. In your class, use whatever your book or instructor uses.) Do you see that the graph (in blue) is shifted over to the right by π/3 units from the regular graph (in gray)? This is always true: If the argument of the function (the thing you're plugging in to the function) is of the form (variable) – (number), then the graph is shifted to the right by that (number) of units; if the argument is of the form (variable) + (number), then the graph is shifted to the left by that (number) of units. So cos(t + π/4) would be shifted to the left by π/4 units, and tan(t – 2π/3) would be shifted to the right by (2/3)π units. This right-or-left shifting is called "phase shift". Do you see how the graph was shifted up by threeunits? This is always true: If a number D is added outside the function, then the graph is shifted up by that number of units; if D is subtracted, then the graph is shifted down by that number of units. So cos(t) – 2 is the regular cosine wave, but shifted downward two units; and tan(t) + 0.6 is the regular tangent curve, but shifted upward by 6/10 of a unit.
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Putting it all together, we have the general sine function, F(t) = Asin(Bt – C) + D, where A is the amplitude, B gives you the period, D gives you the vertical shift (up or down), and C is used to find the phase shift. Why don't you always just use C? Because sometimes more is going on inside the function. Remember that the phase shift comes from what is added or subtracted directly to the variable. So if you have something like sin(2t – π), the phase shift is not π units! Instead, you first have to isolate what's happening to the variable by factoring: sin(2(t – π/2)). Now you can see that the phase shift will be π/2 units, not π units. So the phase shift, as a formula, is found by dividing C by B.
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