To convert from one unit to another, first find a relationship between the units. Then use this relationship to convert from one to the other. A relationship between the units may be obtained by finding the ratio between the two units.
Example: Convert 5 km to mm.
Solution: First we have to find a relationship between km and mm by finding the ratio between km and mm. The unit meter (m) cancels out, so the ratio is basically ratio between kilo (k) and milli (m). Remember k = 10 3 and m = 10 -3.
km ⁄ mm = k ⁄ m = 10 3 ⁄ 10 -3 = 10 6
km = 10 6 mm
5 km = 5 x 10 6 mm
Example: Convert 5 cm 2 to m 2
Solution: First we have to find the ratio between the two units. The unit (m 2) will cancel out and the ratio simplifies to c 2 (The square in cm applies to both c and m). Remember c = 10 -2.
cm 2 ⁄ m 2 = c 2 = 10 -4
cm 2 = 10 -4 m 2
5 cm 2 = 5 x 10-4 m 2
Dimensional Analysis is a method used to determine if an equation is sound or not by comparing the units of both sides of the equation. Both sides of an equation are obviously expected to have the same units. If they turn out to be different, then the equation must be wrong; if they turn out to be the same, then the equation is at least dimensionally correct and there is a good chance it may be correct.
Example: Determine if the equation speed = acceleration x time is dimensionally correct.
Solution: The unit for speed is m ⁄ s. The unit for acceleration is m ⁄ s 2. Thus, the unit for the left side of the equation is m ⁄ s and that of the right hand side is the product of m ⁄ s 2 and s which is equal to m ⁄ s. Since both sides have the same units, the equation is dimensionally correct.
Calculating the order of magnitude of a certain calculation means approximating its power of ten. To obtain the order of magnitude, replace every number by the power of ten closest to it and then carry out the calculation.
Example: Obtain the order of magnitude of the following calculation: 34 * 4 * 786 * 9876
Solution: 34 is between 10 1 and 10 2 and is closer to 10 1. 4 is between 10 0 and 10 1 and is closer to the former. 786 is between 10 2 and 10 3 and is closer to the later. 9876 is between 10 3 and 10 4 and is closer to the later. Therefore the order of magnitude of this calculation is
10 1 * 10 0 * 10 3 * 10 4 = 10 8
There are three basic trigonometric functions which are the cosine, the sine and the tangent. They are defined in terms of a right angled triangle. A right angled triangle is a triangle whose largest angle's measure is 90°. Its longest side or the side opposite to the 90 degree angle is called the hypotenuse. The other two sides are called the legs of the right angled triangle. Now consider one of the none 90° angles. This angle is formed by the hypotenuse and one of the legs. The later is called the adjacent side. The other leg which is not part of this angle is called the opposite side. Let this angle be x. the adjacent side be a, the opposite side be b and the hypotenuse be c.
The cosine of this angle, written as cos x, is defined to be the ratio between the adjacent side and the hypotenuse.
cos x = a ⁄ c
The sine of this angle, written as sin x, is defined to be the ratio between the opposite side and the hypotenuse.
sin x = b ⁄ c
The tangent of this angle, written as tan x , is defined to be the ratio between the opposite side and the opposite side.
tan x = b ⁄ a
The values of trigonometric functions are available in a scientific calculator.
The hypotenuse and the legs of a right angled triangle are related by Pythagorean theorem.
c² = a² + b²
Example: The degree measure of one of the angles of a right angled triangle is 60°. The hypotenuse is 10. Calculate its adjacent side.
Solution: c = 10 ; x = 60° ; a = ?
The trigonometric function that relates these 3 values is cosine.
cos x = a/c
a = c cos x = 10 cos 60° = 5
The inverses of trigonometric functions, called inverse trigonometric functions, help you to recover an angle from the value of a trigonometric function.
Cosine inverse of a trigonometric value a, written as cos-1 a or arccos a, is defined as follows:
If
cos x = a,
then
x = arccos a
Sine inverse of a trigonometric value a, written as sin-1 a or arcsin a, is defined as follows:
If
sin x = a,
then
x = arcsin a
Tangent inverse of a trigonometric value a, written as tan-1 a or arctan a, is defined as follows:
If
tan x = a,
then
x = arctan a
Example: In a certain right angled triangle of hypotenuse 50, the side opposite to one of the angles, x, is 25. Calculate the angle.
Solution: c = 50; b = 25; x = ?
sin x = b ⁄ c = 25 ⁄ 50 = 0.5
x = arcsin 0.5 = 30°
tan x = (sin x)/(cos x)
cos² x +sin² x = 1
sin (x ± y) =sin (x) cos (y) ± cos (x) sin (y)
cos (x ± y) = cos (x) cos (y) ± (-) sin (x) sin (y)
sin (2x) = 2 sin x cos x
cos (2x) = cos² x - sin² x
A coordinate system is a system for associating a set of three numbers with points in space uniquely. Two special cases of this are the one dimensional coordinate system and the two dimensional coordinate system.
Tho one dimensional coordinate system is also called a number line. It is a system that associates single numbers with points on a line uniquely. A point is related with a number that is equal to its distance from a certain point that we call a reference point or origin. To distinguish between distances to the right of the origin and distances to the left of the origin, distances to the right of the origin are taken to be positive while distances to the left of the origin are taken to be negative.
A Two dimensional coordinate system is also called a coordinate plane. It is a system for associating pairs of numbers with points in a plane uniquely. There are two kinds of two dimensional coordinate system which are the Cartesian coordinate system and the polar coordinate system.
In this kind of coordinate system, a point is related to its distance from the origin and the angle formed between the line joining the origin to the point and the positive x-axis. The angle is taken to be positive if measured in a counter clockwise direction from the positive x-axis and negative if measured in a clockwise direction from the positive x-axis. The distance and the angle are customarily represented by r and θ respectively. The coordinate of the point is represented by the ordered pair (r, θ).
Consider the right angled triangle formed by the side joining the origin to the given point and a line extended from the point to the x-axis perpendicularly. The length of the hypotenuse is equal to r. The angle formed between the horizontal leg and the hypotenuse is equal to θ. The lengths of the horizontal and vertical legs are respectively equal to the x and y coordinates of the point (if in the first quadrant). The following relationships between Cartesian and polar coordinates can be obtained easily using the definitions of trigonometric functions.
x = r cos θ
y = r sin θ
r = √(x² + y²)
θ = arctan (y/x)
Example: The polar coordinate of a certain point is ( 100 , 60° ). Calculate its Cartesian coordinates.
Solution: r = 100; θ = 60°; x = ? ; y = ?
x = r cos θ = 100 cos 60° = 50
y = r sin θ = 100 sin 60° = 87
r = √(x² + y²) = √(64² + 48²) = 80
θ = arctan (y ⁄ x) = arctan (48 ⁄ 64) = 37°