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7-4 Evaluating and Graphing Sine and Cosine Objective: To use reference angles, calculators or tables, and special angles to find values of the sine and cosine functions and to sketch the graphs of these functions.

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Let (Greek alpha) be an acute angle in standard position. (Quadrant 1) (x,y) (-x,y) = 20˚ and is the reference angle for 180˚ ± 20˚ and 360˚ - 20˚ 20˚ is the reference angle for 160˚, 200˚, and 340˚, etc Sin 160˚ = sin 20˚ Cos 160˚ = -cos 20˚

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Example 1Find the reference angle for each angle. (a)218 º (b) [Solution] (a) = 218 º – 180 º = 38 º (b) Reference Angles A reference angle for an angle , written , is the positive acute angle made by the terminal side of angle and the x-axis.

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(x, y) = (cos , sin ) y x r = 1 0 (–x, –y) = (cos , sin ) Why do we need to learn the reference angles? sin = – y = – sin cos = – x = – cos Since the values of sine and cosine for an angle is determined by the x and y coordinates from a point of intersection of the terminal side of this angle and the unit circle. From the above observation, we notice that we can transfer evaluating sin and cos by the sine and cosine of the reference angle of , which is . (x, y) = (cos , sin ) y x r = 1 0 (x, –y) = (cos , sin ) sin = – y = – sin cos = x = cos = + = 2 –

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Notice how the acute angle is symmetrical in the y axis. In each case, is the reference angle for

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Express sin 840 in terms of a reference angle. 840/360 = 2 with a remainder of 120. The reference angle for 120 is 180 – 120 = 60. 120 is in quadrant 2 so sin would be +. sin 840 = sin 60

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Common angles and their sine and cosine 0o00o0 30 o 45 o 60 o 90 o sin 01 cos 10 Note In the first quadrant (including x and y-axis), the sine function is increasing and the cosine function is decreasing.

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Example 3 Find the exact value for cos135 o and sin330 o [Solution] Since 90 o < 135 o < 180 o, and 135 o = 180 o – 45 o Then, the reference angle is 45 o. So cos135 o = – cos45 o = Since 270 o < 330 o < 360 o, and 330 o = 360 o – 30 o Then, the reference angle is 30 o. So sin330 o = – sin30 o =

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Find the exact value of each expression. 2x = -1 x = -1/2

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Unit Circle (cosine, sine) Note: Memorize 0˚ through 90˚, then use reference angles to determine the remaining angles.

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Calculator Graphing Specify range parameters for functions

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Sketch the graphs of y= -sin x and y = x/3 on the same set of axes. How many solutions does the equation –sin x = x/3 1

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x y (0, 1) 90° (–1, 0) 180° (0, –1) 270° (1, 0) 0° 60° 45° 30° 330° 315° 300° 120° 135° 150° 210° 225° 240° y 1 – 1 0 Graphs of Sine and Cosine On the circumference of a unit circle, label the special angles in radian, the y coordinates of these special points are the sine values. I) In [0, /2), sine value 0 1 II) In [ /2, ) sine value 1 0 III) In [ , 3 /2), sine value 0 -1 VI) In [3 /2, 2 ) sine value -1 0

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Graphs of Sine and Cosine

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Since the sine function is periodic with a fundamental period of 360 o or 2 , the graph above can be extended left and right as show below.

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To graph the cosine function, we analyze the x coordinate of the rotating particle in a similar manner, since the cosine function has the fundamental period of 360 o or 2 , the graph above can be extended left and right as show below:

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ASSIGNMENT Page 279 2-10 even, 11-18 all, 21-23 all

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From graphs below, we find 1.sine graph and cosine graph have the same shape. 2.each one is the horizontal transformation to the other. 3.sine graph is symmetry to the origin and therefore it is an odd function. sin(– )= – sin 4.cosine graph is symmetry to the y-axis and therefore it is an even function. cos(– )= cos 5.sin = cos( – /2) or sin = cos( – 90 o ) 6.cos = cos( + /2) or sin = cos( + 90 o )

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Simple Trigonometric Equations The sine graph below illustrates that there are many solutions to the trigonometric equation sin x = 0.5.

Simple Trigonometric Equations The sine graph below illustrates that there are many solutions to the trigonometric equation sin x = 0.5.

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