Functions+Notes+_updated_.pdf

Cos 2 θ sin 2 θ 1 3 periodicity since c has

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cos 2 θ + sin 2 θ = 1 3. Periodicity. Since C has circumference 2 π , adding 2 π to θ causes the point P = (cos θ, sin θ ) to go one extra complete revolution around C
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2. Functions 53 and end up in the same place. In other words, the angles θ and θ + 2 π have the same terminal side. Thus: sin( θ + 2 π ) = sin θ cos( θ + 2 π ) = cos θ These identities show that the sine and cosine functions are peri- odic with period 2 π . 4. Cosine is an even function. Sine is an odd function. It follows from the fact that the circle x 2 + y 2 = 1 is symmetric about the x - axis, so the points (cos θ, sin θ ) and (cos( - θ ) , sin( - θ )) have the same x -coordinates and opposite y -coordinates. sin( - θ ) = - sin θ cos( - θ ) = cos θ 5. Addition formulas sin( x + y ) = sin x cos y + cos x sin y cos( x + y ) = cos x cos y - sin x sin y sin( x - y ) = sin x cos y - cos x sin y cos( x - y ) = cos x cos y + sin x sin y Dividing the formulas above we get tan( x + y ) = tan x + tan y 1 - tan x tan y tan( x - y ) = tan x - tan y 1 + tan x tan y 6. Double-angle formulas. If we put x = y in the addition formulas we get sin(2 x ) = 2 sin x cos x cos(2 x ) = cos 2 x - sin 2 x Using that cos 2 + sin 2 = 1 we also have the following alternate forms of the double-angle formulas for cos(2 x ): cos(2 x ) = 2 cos x 2 - 1 cos(2 x ) = 1 - 2 sin 2 x
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54 J. S´ anchez-Ortega 7. Half-angle formulas. If we now solve these equations for cos 2 and sin 2 , we get the following formulas, which are useful in integral calculus: cos 2 x = 1 + cos 2 x 2 sin 2 x = 1 - cos 2 x 2 8. Other consequences of the addition formulas are: cos( θ + π ) = cos( θ - π ) = - cos θ sin( θ + π ) = sin( θ - π ) = sin θ tan( θ + π ) = tan( θ - π ) = tan θ sin θ + π 2 = cos θ cos θ - π 2 = sin θ The following table summarizes the values of cosine and sine at some common angles. In what follows, we will apply the previous identities to find the values of cosine and sine at certain angles. Example 2.41. cos 4 π 3 = cos π + π 3 = cos( π ) cos π 3 - sin( π ) sin π 3 = - 1 2
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2. Functions 55 cos π 12 = cos π 3 - π 4 = cos π 3 cos π 4 + sin π 3 sin π 4 = = 1 2 1 2 + 3 2 1 2 ! = 1 + 3 2 2 = 2(1 + 3) 4 sin 7 π 6 = sin π + π 6 = sin π 6 = 1 2 Remember this! When using a scientific calculator to calculate any trigono- metric functions, be sure you have selected the proper angular mode: degrees or radians. Graphs of the Trigonometric Functions The graph of the function f ( x ) = sin x can be obtained by plotting points for 0 x 2 π and then using the fact that it is a periodic function to periodic to complete the graph. Notice that the zeros of the sine function occur at the integer multiples of π , that is sin x = 0 whenever x = nπ, n an integer Because of the identity cos x = sin x + π 2 the graph of cosine is obtained by shifting the graph of sine π/ 2 units to the left.
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56 J. S´ anchez-Ortega Notice that for both the sine and cosine functions the domain is R and the range is the closed interval [ - 1 , 1]. Thus, for all values of x , we have - 1 sin x 1 - 1 cos x 1 The graph of the tangent is as follows: Note that tangent has range R and domain R - π 2 + kπ, k Z . It is a periodic function of period π .
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