$geometry calculator sin cos tan$

Some rules/guidelines for trigonometry of right triangles The side opposite the right angle, H, is always the longest side and is called the hypotenuse. The side 'opposite' an angle (in this case) is labeled with a capital letter corresponding to the label on the angle. Note that the triangle on the right has 3 angles a, b and c and 3 sides, A, B, and H, and 3 angles a, b, and c. The other two angles are always less than 90 В° and together add up to 90 В°. It is important to understand the relationships between the sine and cosine functions and how phase shifts can alter their graphs.A right triangle (like the one in the figure to the right) has one angle that is 90 В°. Thus, Figure can also represent the graph of y = в€’3 sin 2 x. The graph of the function y = в€’ f( x) is found by reflecting the graph of the function y = f( x) about the xвЂђaxis. In other words,Д®xamples of several phase shifts of the sine function.Д®xample 3: What is the amplitude, period, phase shift, maximum, and minimum values ofД®xample 4: Sketch the graph of y = cosПЂ x.Д«ecause cos x has a period of 2ПЂ, cos ПЂ x has a period of 2 (Figure 9).Д®xample 5: Sketch the graph of y = 3 cos (2x + ПЂ/2).Д«ecause cos x has a period of 2ПЂ, cos 2x has a period of ПЂ (Figure 10). The cosine function looks exactly like the sine function, except that it is shifted ПЂ/2 units to the left (Figure 8). The sine function is odd, and the cosine function is even. It does not matter whether the shift is to the left (if D is positive) or to the right (if D is negative). (This also holds for the cosine function.) The phase shift is | D |. The additional term D in the function y = sin ( x + D) allows for a phase shift (moving the graph to the left or right) in the graph of the sine functions. Figure 7В illustrates additional examples.Д®xamples of several frequencies of the a) sine function and b) cosine function. Thus, the function y = sin 5 x has a period of 2ПЂ/5. (This also holds for the cosine function.) The period of the function y = sin Cx is 2ПЂ/|C|.

The additional factor C in the function y = sin Cx allows for period variation (length of cycle) of the sine function. The minimum value is 4 в€’ 3 = 1 (Figure 6В ). What are the maximum and minimum values of the function? The minimum value is 1 в€’2 = в€’1 (Figure 5В ).Д®xample 2: Graph the function y = 4 + 3 sin x. This minimum occurs whenever sin x = в€’1 or cos x = в€’1.Д®xample 1: Graph the function y = 1 + 2 sin x. The minimum value of the function is m = A вЂђ |B|.

This maximum value occurs whenever sin x = 1 or cos x = 1. The maximum value of the function is M = A + |B|. These two functions have minimum and maximum values as defined by the following formulas. This also holds for the cosine function (Figure 4В ).Д®xamples of several amplitudes of the sine function.Д¬ombining these figures yields the functions y = A + B sin x and also y = A + B cos x. The amplitude, | B |, is the maximum deviation from the xвЂђaxis-that is, one half the difference between the maximum and minimum values of the graph. The additional factor B in the function y = B sin x allows for amplitude variation of the sine function. This also holds for the cosine function (Figure 3В ).Д®xamples of several vertical shifts of the sine function. The additional term A in the function y = A + sin x allows for a vertical shift in the graph of the sine functions. Several additional terms and factors can be added to the sine and cosine functions, which modify their shapes. Multiple periods of the a) sine function and b) cosine function. The sine function and the cosine function have periods of 2ПЂ therefore, the patterns illustrated in Figure are repeated to the left and right continuously (Figure 2В ). One period of the a) sine function and b) cosine function. Next, plot these values and obtain the basic graphs of the sine and cosine function (Figure 1В ). To see how the sine and cosine functions are graphed, use a calculator, a computer, or a set of trigonometry tables to determine the values of the sine and cosine functions for a number of different degree (or radian) measures (see Table 1).

Graphs: Special Trigonometric Functions.

Geometry calculator sin cos tan