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Trigonometry Lecture Notes_part3

Step 1 use the law of cosines to find the angle

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Step 1: Use the Law of Cosines to find the angle opposite the longest side. 2 2 2 2 cos b a c ac B = + - Solve for cosB 2 2 2 2 a c b cosB ac + - = Enter your given side values 29 cos 48 B = - Since the cosine is negative, B is obtuse 1 29 cos 127.2 48 B - = - Step 2: Apply the Law of Sines Step 3: Find the third angle by subtraction. Example 101 Applying Law of Cosines Two airplanes leave an airport at the same time on different runways. One flies at a bearing of N66ºW at 325 miles per hour. The other airplane flies at a bearing of S26ºW at 300 miles per hour. How far apart will the airplanes be after two hours?
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The area of a triangle with sides a , b , and c is Example 102 Using Heron’s Formula Use Heron’s formula to find the area of the given triangle: a = 10m, b = 8m, c = 4m ( )( )( ) 1 ( ) 2 Area s s a s b s c s a b c = - - - = + + 1 ( ) 2 1 (10 8 4) 2 1 (22) 11 2 s a b c s s = + + = + + = = ( )( )( ) 11(11 10)(11 8)(11 4) 11(1)(3)(7) 231 . . Area s s a s b s c sq m = - - - = - - - = =
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Section 11.5 Polar Coordinates The Sign of r and a Point’s Location in Polar Coordinates: The point P = ( r , θ ) is located | r | units from the pole. If r > 0, the point lies on the terminal side of θ . If r < 0 the point lies along the ray opposite the terminal side of θ . If r = 0 the point lies at the pole, regardless of the value of θ . Example 103 Plot the points with the following polar coordinates: a. (2, 135°)
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a. 3 3, 2 π - c. 1, 4 π - - Multiple Representations of Points In the rectangular coordinate system a point is uniquely represented by its x and y coordinates; however, this is not true for polar points. They have many representations: If n is any integer, the point ( r , θ ) can be represented as ( r , θ ) = ( r , θ + 2 n π ) or ( r , θ ) = (- r , θ + π + 2 n π ) Example 104 Find another representation of 5, 4 π in which: a. r > 0 and 2 4 π θ π < < b. r < 0 and 0 2 θ π < < Relations between Polar and Rectangular Coordinates Example 105 Find the rectangular coordinates for the following polar points: a. ( ) 3, π b. 10, 6 π -
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Converting a Point from Rectangular to Polar Coordinates (r > 0 and 0 < θ < 2 π ) 1. Plot the point (x, y). 2. Find r by computing the distance from the origin to (x, y). 3. Find θ using tan θ = y/x with θ lying in the same quadrant as (x, y).
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