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A Mathematics Review Unit 1 Presentation 2 Why Review? Mathematics are a very important part of Physics Graphing, Trigonometry, and Algebraic concepts are used often Solving equations and breaking down vectors are two important skills Graphing Review Graphing in Physics done on a Cartesian Coordinate System Also known as an x-y plane Can also graph in Polar Coordinates Also known as an r,q plane Very Useful in Vector Analysis Rectangular vs. Polar Coordinates Rectangular Coordinate System X-Y Axes Present (dark black lines) X Variable Y Variable Polar Coordinate System NO X-Y Axes R Variable (red lines) q Variable (blue/black lines) Trigonometry Review Remember SOHCAHTOA? Opposite Side opposite hypotenuse adjacent cos q hypotenuse opposite sin q tan q adjacent cos q sin q Pythagorean Theorem for Right Triangles: a b c 2 q Adjacent Side 2 2 Using Polar Coordinates To convert from Rectangular to Polar Coordinates (or vice versa), use the following: r x y 2 2 y q tan x 1 x r cos q y r sin q Polar Coordinates Example Convert (-3.50 m, -2.50 m) from Cartesian coordinates to Polar coordinates. x 3.50m r x 2 y 2 r (3.50) 2 (2.50) 2 4.30m y 2.50m 1 y q tan 1 2.50 36 x q tan 3.50 But, consider a displacement in the negative x and y directions. That is in Quadrant III, so, since polar coordinates start with the Positive x axis, we must add 180° to our answer, giving us a final answer of 216° Another Polar Coordinates Example Convert 12m @ 75 degrees into x and y coordinates. First, consider that this displacement is in Quadrant I, so our answers for x and y should both be positive. r 12m q 75 x r cosq x 12 cos(75) 3.11m y r sin q y 12 sin( 75) 11.60m Trigonometry Review Calculate the height of a building if you can see the top of the building at an angle of 39.0° and 46.0 m away from its base. 39.0° Building Height First, draw a picture. Since we know the adjacent side and want to find the opposite side, we should use the tangent ratio. 46.0 m q 39.0 adjacent 46.0m opposite ? tan q opposite adjacent tan q adjacent opposite tan( 39.0) 46.0m 37.3m Another Trigonometry Example An airplane travels 4.50 x 102 km due east and then travels an unknown distance due north. Finally, it returns to its starting point by traveling a distance of 525 km. How far did the airplane travel in the northerly direction? First, draw a picture. x km N 450 km This problem would best be solved using the Pythagorean Theorem. a2 b2 c2 a 450km b? b c2 a2 c 525km b (525) 2 (450) 2 270km