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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

cm)2 102

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 C c a sin C c (40 cm) (sin 600 ) 46

The inverses of sine, cosine, and tangent allow you to do the reverse of the sine, cosine, and tangent functions and find the angle The trigonometric functions and their inverses are as follows: Trigonometric Function y y y sin x cos x tan x x x x sin cos tan

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@interface WhatAboutThatFileAppDelegate : NSObject { NSFileWrapper *fileWrapper; NSString *filePath; NSStringEncoding chosenEncoding; } @property (retain) NSFileWrapper *fileWrapper; @property (retain) NSString *filePath; @property (readonly) NSDictionary *fileAttributes; @property (readonly) NSString *filename; @property (readonly) NSImage *fileIcon; @property (readonly) NSImage *opensAppIcon; @property (readonly) NSString *opensAppName; @property (assign) NSString *stringEncodingName; @property (readonly) NSString *fileStringValue; @property (readonly) NSDictionary *encodingNames; @property (assign) NSStringEncoding chosenEncoding; - (IBAction)chooseFile:(id)sender; @end

The sine function, y sin x, and the cosine function, y cos x are periodic functions The period for each function is 2 x can be any real number y is any real numbers between 1 and 1, inclusive

Let b and x be positive integers, b 1 The logarithm of x with base b, written logb x, equals y, where y is the exponent that makes the equation by x true The log to the base b of x is the number to which you can raise b to get x logb x y if and only if b y x Memory aid: the log is the exponent Examples: Calculate the following logarithms log2

1 16

1 16

When you want to find the log of a number, you also can use a calculator Connecting Math to Physics Physicists use logarithms to work with measurements that extend over several orders of magnitude, or powers of 10 Geophysicists use the Richter scale, a logarithmic scale that allows them to rate earthquakes from 0 to 7, or larger, though the power of earthquakes differ by 7 or more powers of 10

test=> select "CustomerID" as "ID", "LastName" as "Family", test-> "FirstName" as "Person" from store"Customer"; ID | Family | Person + + BLU004 | Blum | Jessica BLU001 | Blum | Rich BLU002 | Blum | Barbara BLU003 | Blum | Katie (4 rows) test=>

Base 10 logarithms are called common logarithms They often are written without the subscript 10 log10 x log x x 0

Now let s move on to the .m file:

An antilogarithm is the inverse of a logarithm An antilogarithm is the number which has a given logarithm Example: Solve log x log x x 4 10 4 4 for x

Connecting Math to Physics An equation for loudness, L, in decibels, is L 10 log10 R, where R is the relative intensity of the sound Calculate R for a fireworks display with a loudness of 130 decibels 130 13 R 10 log10 R log10 R 1013

When you know the logarithm of a number and want to know the number itself, use a calculator to find the antilogarithm

4 0001

1

#import "WhatAboutThatFileAppDelegate.h" @implementation WhatAboutThatFileAppDelegate @synthesize fileWrapper, filePath, chosenEncoding; - (void)applicationDidFinishLaunching:(NSNotification *)n { fileWrapper = nil; filePath = nil; }

1 The density, , of an object is given by the ratio of the object s mass, m, and volume, V, according to the equation m/V What is the density of a cube that is 12 cm on each side and has a mass of 256 g 2 An object that is moving in a straight line with speed v covers a distance, d vt, in time t Rewrite the equation to find t in terms of d and v How long does it take a plane that is traveling at 350 km/h to travel 1750 km 3 Convert 523 kg to milligrams 4 The liquid measure milliliter, mL, is the same as 1 cm3 How many milliliters of liquid can be held in a 25-m3 container 5 Part of the label from a vitamin container is shown below The abbreviation mcg stands for micrograms Convert the values to milligrams

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