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Inverses of Sine, Cosine, and Tangent
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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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1y 1y 1x
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Inverse or x or x or x arc sin y arc cos y arc tan y
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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
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y sin x
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Appendix A
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Math Handbook
IX Logarithms
Math Handbook
Logarithms with Base b
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
Because 2
log10 1000
Because 10 3
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
Common Logarithms
Base 10 logarithms are called common logarithms They often are written without the subscript 10 log10 x log x x 0
Antilogarithms or Inverse Logarithms
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
104 is the antilogarithm of 4
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
Divide by 10 Use logarithm rule
When you know the logarithm of a number and want to know the number itself, use a calculator to find the antilogarithm
20 Write in exponential form log3 81 21 Write in logarithmic form 10 22 Find x log x
4 0001
Math Handbook
1
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
7 How many significant digits are there in each of the following measurements a 100 m b 00023 m/s c 1001 m d 20023 8 The buoyant (upward) force exerted by water on a submerged object is given by Vg, where is the the formula F density of water (10 103 kg/m3), V is the volume of the object in m3, and g is the acceleration due to gravity (980 m/s2) The force is in newtons Rewrite the equation to find V in terms of F Use this to find the volume of a submerged barrel if the buoyant force on it is 9200 N 9 What is 23 kg 023 g 10 Solve the following problems a 155 cm 121 cm b
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