Trigonometry Lecture Notes_part3

# Example 106 find the polar coordinates of a point

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Example 106 Find the polar coordinates of a point whose rectangular coordinates are (2, 4) Example 107 Find the polar coordinates of a point whose rectangular coordinates are (0, -4) Example 108 Converting an equation from Rectangular to Polar Coordinates Convert 2x-y=1 to a polar equation.

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Converting Equations from Polar to Rectangular Form Recall: We will use the above relationships to rewrite polar equations into rectangular form. Example 109 Convert each polar equation to a rectangular equation in x and y: a. r = 4 b. 3 4 π θ = c. sec r θ = Section 11.5 Graphs of Polar Equations Using Polar Grids to Graph Polar Equations Recall that a polar equation is an equation whose variables are r and è . The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation. We use polar grids like the one shown to graph polar equations. The grid consists of circles with centers at the pole. This polar grid shows five such circles. A polar grid also shows lines passing through the pole, In this grid, each fine represents an angle for which we know the exact values of the trigonometric functions.
One method of graphing polar equations is to use point plotting. We will create a table of values just as we do with graphs in x and y. Example 110 Graph the polar equation r = 4 cos θ with θ in radians.

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Testing for Symmetry in Polar Coordinates (failure does not indicate a lack of symm.) To test or symmetry with respect to the x-axis, replace θ with θ - . To test or symmetry with respect to the y-axis, replace ( ) , r θ with ( ) , r θ - - . To test or symmetry with respect to the origin, replace r with r - . Example 111 Check for symmetry and then graph the polar equation: r = 1 - cos θ .

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Example 112 Graph 1 2sin r θ = + (use symmetry to assist you)
Example 113 Graph the polar equation y= 2+3cos θ

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Example 114 Graph the polar equation y=3sin2 θ
Example 115 Graph 2 4sin 2 r θ =

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Section 8.6 Complex Numbers in Polar Form; DeMoivre’s Theorem Example 116
Example 117 Determine the absolute value of of each of the following complex numbers: a. 5 12 z i = + b. 2 3 z i = - Example 118 Determine the absolute value of z=2-4i 2 2 2 2 2 ( 4) 4 16 20 2 5 z a bi a b = + = + = + - = + = =

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Example 119
r = a 2 + b 2 = ( - 2) 2 + ( - 2) 2 = 4 + 4 = 8 = 2 2 tan θ = b a = - 2 - 2 = 1 z = r (cos θ + i sin θ ) = 2 2 (cos 5 π 4 + i sin 5 π 4 ) Example 120 Writing a complex number in rectangular form: Write ( ) 4 cos30 sin30 z i = °+ ° in rectangular form.

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