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Time (s) 1 2 3 4 Position (m) 3 6 11 18
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Position (m) 20 16 12 8 4 0 2 4 Quadratic Graph of Constant Acceleration
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Time (s)
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Appendix A
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Math Handbook
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VIII Geometry and Trigonometry
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Perimeter, Area, and Volume
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Perimeter, Circumference Linear units Square, side a Rectangle, length l width w Triangle base b height h Cube side a Circle radius r Cylinder radius r height h Sphere radius r C 2 r A r2 P 4a Area Squared units Surface Area Squared units Volume Cubic units
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1 bh 2
6a 2
2 rh
2 r2
r 2h
4 3 r 3
4 r2
Connecting Math to Physics Look for geometric shapes in your physics problems They could be in the form of objects or spaces For example, two-dimensional shapes could be formed by velocity vectors, as well as position vectors
Area Under a Graph
To calculate the approximate area under a graph, cut the area into smaller pieces and find the area of each piece using the formulas shown above To approximate the area under a line, cut the area into a rectangle and a triangle, as shown in (a) To approximate the area under a curve, draw several rectangles from the x-axis to the curve, as in (b) Using more rectangles with a smaller base will provide a closer approximation of the area
Position
Position Time Total area Area of the rectangle Area of the triangle 0 Time Total area Area 1 Area 2 Area 3 Math Handbook
Right Triangles
The Pythagorean theorem states that if a and b are the measures of the legs of a right triangle and c is the measure of the hypotenuse, then c 2 a 2 b 2 To determine the length of the hypotenuse, use the square root property Because distance is positive, the negative value does not have meaning c a2 b2 4 cm and b 3 cm Find c
Math Handbook
Hypotenuse c Leg a
b Leg
Example: In the triangle, a c a2 b2 (4 cm)2 16 cm2 25 cm2 5 cm
(3 cm)2 9 cm2
45 -45 -90 triangles The length of the hypotenuse is 2 times the length of a leg
45 - 45 - 90 Right Triangle 45
( 2 )x
45 x
30 -60 -90 triangles The length of the hypotenuse is twice the length of the shorter leg The length of the longer leg is 3 times the length of the shorter leg
30 - 60 - 90 Right Triangle 2x
60 x
( 3 )x
Appendix A Math Handbook
Trigonometric Ratios
Math Handbook
A trigonometric ratio is a ratio of the lengths of sides of a right triangle The most common trigonometric ratios are sine, cosine, and tangentTo memorize these ratios, learn the acronym SOH-CAH-TOA SOH stands for Sine, Opposite, Hypotenuse CAH stands for Cosine, Adjacent, Hypotenuse TOA stands for Tangent, Opposite, Adjacent
Words The sine is the ratio of the length of the side opposite to the angle over the length of the hypotenuse The cosine is the ratio of the length of the side adjacent to the angle over the length of the hypotenuse The tangent is the ratio of the length of the side opposite to the angle over the length of the side adjacent to the angle Memory Aid SOH sin CAH cos TOA tan
opposite adjacent adjacent hypotenuse opposite hypotenuse
Symbols
Example: In right triangle ABC, if a 3 cm, b 4 cm, and c 5 cm, find sin and cos sin cos
3 cm 5 cm 4 cm 5 cm
06 cm 08 cm 300 and
b 200 cm
A b Side adjacent Hypotenuse c
Example: In right triangle ABC, if c 200 cm, find a and b sin 300 a b
a 200 cm
Side opposite
cos 300 100 173
(200 cm)(sin 300 ) (200 cm)(cos 300 )
Law of Cosines and Law of Sines
The laws of cosines and sines let you calculate sides and angles in any triangle Law of cosines The law of cosines looks like the Pythagorean theorem, except for the last term is the angle opposite side c If the angle is 90 , the 0 and the last term equals zero If is cos greater than 90 , its cosine is a negative number c 2 a 2 b 2 2ab cos Example: Find the length of the third side of a 1100 triangle with a 100 cm, b 120 cm, c2 c a2 a2 (100 100 163
Math Handbook
b2 b2
2ab cos 2ab cos (120 cm)2 144 cm2 2(100 cm)(120 cm)(cos 1100 ) (600 cm2 )(cos 1100 ) cm2
cm)2 102
Math Handbook
Law of sines The law of sines is an equation of three ratios, where a, b, and c are the sides opposite angles A, B, and C, respectively Use the law of sines when you know the measures of two angles and any side of a triangle
sin A a sin B b sin C c
C b A
Example: In a triangle, C 600 , a 40 cm, c 46 cm Find the measure of angle A
sin A a sin C c a sin C c (40 cm) (sin 600 ) 46
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