Chapter 1: Algebra and Geometry Review1.1 AlgebraAlgebra is a part of mathematics which solves problems by representing quantities by symbols (often called variables), expressing the relationships between the quantities as equations or inequalities, and manipulating these expressions according to well defined rules in order to find additional properties of the quantities and solve the problem. This course course assumes that you have already had considerable experience in using algebra, and that the material in this chapter is, for the most part, simply review. The remainder of this section will deal with rules for manipulating algebraic expressions, for solving linear and quadratic equations in one unknown, for solving systems of equations in more than one unknown, and for solving inequalities. 1.1.1 Simplifying ExpressionsWe begin with a review of the basic rules for simplifying algebraic expressions. You probably already know hundreds of these, and the purpose here is point out those that we think are the most important. This will reduce the number of such rules down to a few dozen; in subsequent chapters, we will see that even this short list can be pared down to a few fundamentals from which all the rest follow. Some really fundamental rules are:
The following examples show how these three fundamental rules can be combined to simplify more complicated expressions: Example 1: You can distribute over longer sums by applying the distributive law multiple times:
Example 2: One can multiply out products of arbitrary sums using the Example 1:
Operator Precedence: In the above expressions we have been omitting parentheses. For example, we write ab + ac which might be interpreted as a(b + a)c, but we know that the intention is (ab) + (bd). We know this because of the following precedence rules:
Note: Depending on the context, multiple exponentiations without parentheses might be interpreted as being grouped left to right, righjt to left, or as simply being syntactically incorrect. The best practice is to always use an explicit parenthesization to avoid any possible misinterpretation. Example 3: Be careful to evaluate operators of equal precedence from left to right. For example, and not 3/8. Negatives: These are the main properties of the negation operator (unary minus):
Fractions: These are the main properties of fractions. In all the formulas, one assumes that the quantities in the denominators are all non-zero.
Powers: One can raise arbitrary real numbers to integer powers. One defines and, by induction, . For negative integers , one defines . For non-negative real numbers and positive integers , one can define the root to be unique real number such that . This allows one to define rational powers of non-negative numbers by . The following properties are true for these rational powers:
1.1.2 Solving EquationsThe last section was concerned with the problem: Given the values of some quantities, how do you calculate expressions involving those quantities?Most of the course will be concerned with the inverse problem: Given the value of some expressions involving quantities, how do you find the values of the quantities. The two principal arithmetic operations are addition and multiplication. Here is a problem of the second type: Problem 1: Find two numbers given the values of their sum and product. The solution of this problem was already known to the Babylonians, and is one of the most important algebra problems known to them. How can we solve it? First let use represent the two numbers by the letters x and y and represent their sum and product by the letters a and b. The problem can then be expressed as: Given a and b, find x and y such that
Now there are lots of pairs x and y such that x + y = a. One possibility is to take two equal numbers, this would give x = y and our equation becomes 2x = 2y = a. So, x = y = a/2. Now, if , we would have our solution. On the other hand, if this weren't the case, we would not have the solution. This would be the case where the two numbers x and y are not equal. We can think of this is x is different from a/2 and so x = a/2 + t for some number t. Since we still want x + y = a, increasing x by t means that we have to decrease y by the same amount, i.e. y = a/2 - t. Now, let's try this as the solution by putting it in the second equation:
Although it was not clear before how to choose the exact value for t, this last equality tells us that and so . We know the square of the number t we need, and so . But then and . Another solution is obtained by letting t be its second possible value, this simply switches the values of x and y. Problem 1 is the most important algebra problem solved in antiquity. The approach we have taken is that of Diophantus of Alexandria. The presentation was very quick; so let us go back and comment on a number of important points:
1.1.3 Solving Equations in one VariableSolving systems of equations was the topic of the last section. Let's concentrate in this section on the special case in which there is only one equation and only one variable. The next section will return to the more general case. There are two basic ideas in working with equations:
Proposition 1: For all real numbers , , and :
Example 4: To solve the general linear equation ax + b = c, we apply the first principle. Assuming that both sides are equal, we can add -b to both sides to get (ax + b) + (-b) = c + (-b) or ax = c - b. If a is non-zero, then one can then multiply both sides by to obtain , which simplifies to . Doing the same operations with numbers, one can start with 2x + 3 = 5. Assuming that both sides are equal, we can add -3 to both sides to get (2x + 3) + (-3) = 5 + (-3) or 2x = 5 - 3. Since the coefficient 2 is non-zero, one can multiply both sides by to obtain , or . Now, substitute 1 for x in the original equation to check to see that it is indeed a solution. If you want to solve: for x, then, assuming that both sides of the equations are equal, one can multiply them by to get So, the solution is x = 1. Again, one should substitute this back into the original equation to check that it is indeed a solution. In each example, we started out by assuming that we had a solution, solved to find out what the solution must have been, and then checked to see that the value actually worked. This last step is NOT just a check for errors in algebra, but is a NECESSARY step in the procedure. For example, suppose you want to solve: . Assuming that you have a solution, we can proceed exactly as in the last example: So, the solution can only be x = 2. But, when you try to substitute this value back into the original equation, you see that the denominator is zero. So, x = 2 is NOT a solution. This means that there are NO solutions to the original equation. Example 5: Suppose you want to solve . Assume that x is a solution to the equation. The left side can be factored to get . Using the second part of Proposition 1, we know that either x - 2 = 0 or x - 4 = 0. So, x = 2 or x = 4. Substituting each of these values back into the original equation verifies that each of two possibilities is indeed a solution of the original equation. Quadratic Formula: Suppose we want to solve the general quadratic equation: . If a = 0, then the equation is bx + c = 0. This is a simple linear equation. If b is not zero, then it has a single solution x = -c/b. If b = 0 and c is not zero, then there are no solutions. Finally, if both b and c are zero, then every real number x satisfies the equation. If a is not zero, then we can divide through by a to put the equation in the form . This is of the form of Problem 2 of the last section. The solutions obtained there were: (provided that all the operations make sense, i.e. a is non-zero and ). Completing the square: This is another approach to solving a quadratic equation, and it is a method we will use in many other problems as well. Let us assume that we wish to solve the equation . If we could rewrite the equation in the form , then it would be easy to solve for x. This can be accomplished by the operation of completing the square. To see how to do it, just expand out this last expression to get . If this is to be the same as , then corresponding coefficients in the two equations must be equal, i.e. we must have: But this system of equation is easy to solve for d and e. We have and . Let's rework our last example using this method: Example 5: Solve . We want to express this in the form . To do this, we take d = a/2 = -6/2 = -3. So, we would get . Rather than trying to remember the formula for e, we can just multiply out the square to get . Comparing the constant terms, we need 9 + e = 8 and so e = -1, i.e. . We can now solve for x, we get and so . So and so the solutions are x = 4 or x = 2. It is easy to verify that these are both solutions of the original equation. 1.1.4 Solving Systems of EquationsWhen there is more than one variable, one often has more than one equation which the values of the variables are required to simultaneously satisfy. For example, in completing the square, we needed to find all solutions of the system of equations: and where the variables were d and e, and a, b, and c were constants. We were not interested in pairs of numbers (d, e) which satisfied just one of the two equations; they needed to satisfy both equations. Another example is the general system of 2 linear equations in two unknowns x and y: ax + by = e, cx + dy = f. Now, in the special case of the system in the last paragraph, it was easy to solve for the variables because one of the equations involved only one of the variables. We could use it to solve for that variable. Having its value, we could substitute its value into the other equation and obtain an equation involving only the second variable; this equation was then solved and we had all the possible solutions of the system. What makes the general system of 2 linear equations look more difficult is that both equations involve both variables. There are two approaches:
Example 6: Consider the system: 2x + 3y = 5, 4x - 7y = -3. Assume that x and y are satisfy both equations. One can proceed using either method:
Example 7: Find all the solutions of the system of equations: , . Assume that one has a solution x, y. Solving the second equation for y, one gets a value y = 1 -x which when substituted into the firs equation gives: or . Collecting terms, we get a quadratic . Factoring and using the second part of Proposition 1, gives x = 0 or x = 1. Substituting these values into our expression for y, gives two possible solutions (x,y) = (0, 1) and (x,y) = (1, 0). Substituting each of these into the original equations, verifies that both of these pairs are solutions of the original system of equations. If we have more than two variables and more equations, we can apply the same basic strategies. For example, if you have three linear equations in three unknowns, you can use one of them to solve for one variable in terms of the other two. Substituting this expression into the two remaining equations gives two equations in two unknowns. This system can be solved by the method we just described. Then the solutions can be substituted back into the expression for the first variable to find all possible solutions. When you have these, substitute each triple of numbers into the original equations to see which of the possibilities are really solutions. One can also use the second approach as is illustrated by the next example. Example 8: Solve the system of equations: x + y + z = 0, x + 2y + 2z = 2, x - 2y + 2z = 4. Assume that (x, y, z) is a solution. Subtracting the first equation from each of the other two equations gives y + z = 2, -3y + z = 4. Now subtracting the first of these from the second gives -4y = 2 or y = -1/2. Substituting this into the y + z = 2 gives z = 5/2. Finally, substituting these into the first of the original equations gives x = -2. So the only possible solution is (x, y, z) = (-2, -1/2, 5/2). Substituting these values into the original equations shows that this possible solution is, in fact, a solution of the original system. 1.1.5 InequalitiesIn addition to the four basic algebraic operations on real numbers, there is also an order relation. The basic properties are: Proposition 2: Let a, b, and c be real numbers.
Just as Proposition 1 allowed one to operate with equations, Proposition 2 allows one to work with inequalities. The main difference is that multiplication tends to complicate things as there are two cases depending on whether the multiplier is positive or negative. Example 9: Solve the general linear inequality ax + b < 0. Using Proposition 2, this implies ax < - b. If a > 0, Proposition 2 gives x < -b/a. On the other hand, if a < 0, then x > -b/a. Finally, if a = 0, then there are no possible solutions if or every real number is a solution if and . We still need to check that our possible solutions are indeed solutions. This is somewhat more difficult because we do not simply have a small number of values of substitute:
The absolute value |r| of a real number r is defined to be r or -r depending on whether or not r is non-negative or not. For example, |2| = 2, but |-2| = -(-2) = 2. Example 10: Find all solutions of |2x - 3| > 5. There are two cases:
1.2 GeometryGeometry is the mathematical study of the relationships between collections of points, curves, angles, surfaces and solid objects including the measurement of these objects and the distances between them. This review will discuss the correspondence between the real numbers and the points of a line, the general idea of analytic geometry, as well as the equations of lines, circles, and parabolas as well as the measurement of distance between points. 1.2.1 The Pythagorean TheoremThe most important result of classical synthetic geometry is: Theorem 1: If is a right triangle with hypotenuse of length c and legs of length a and b, then . To understand its proof, we need to know Proposition 3: The sum of the angles of any triangle is 180 degrees. Proof: Let be an arbitrary triangle. Construct a line DE parallel to the base BC and through the third vertex A as shown in the diagram below. Since BA is transversal to the pair of parallel lines BC and DE, one has . Similarly, CA is transversal to the pair of parallel lines BC and DE. So, . Since DAE is a straight line, we have . From the diagram, we see that as was to be shown. The proof of the Pythagorean Theorem will also require some simple facts about the areas of squares and triangles. For future reference, let's state these and other results as Proposition 4:(Areas and Perimeters)
Now it is easy to see why the Pythagorean Theorem is true. Start with an arbitrary right triangle where C is the right angle and the opposite side (the hypotenuse) is of length c. Let a and b be the lengths of the legs which are opposite to the angles A and B respectively.
There are at least hundreds of extant proofs of the Pythagorean Theorem. The proof given here is quite different from the proof appearing in Euclid's Elements. Corollary 1: i. The length of the diagonal of a rectangle with sides of lengths a and b is precisely . ii. Let T be a triangle with sides of length a, b, and c opposite angles A, B, and C respectively. Then T is a right triangle with right angle C if . Proof: i. The first assertion is an obvious consequence of Theorem 1. ii. Let T' be a triangle where angle C' is a right angle and the sides opposite angles A' and B' are of length a and b respectively. By Theorem 1, the length of the side opposite angle C' is of length . Then corresponding sides of the triangles T and T' are of equal length; so the two triangles are congruent. But then angle C must be equal to angle C' and so angle C is a right angle. We will need to know about similar triangles. Two triangles and are defined to be similar if , , and . By Proposition 3, if two of these equations hold, then so does the third. Proposition 5: If the two triangles and are similar, then the ratio of the lengths of a pair of sides of one triangle is equal to the ratio of the lengths of the corresponding pair of sides of the other triangle, e.g. . 1.2.2 Analytic GeometryThe rough idea of analytic geometry is to model the plane as the set of all pairs of real numbers. A curve is then the set of solutions of some algebraic equation. In order to solve a geometric problem, one first translates it into an algebra problem about the sets of algebraic equations. Then one uses algebra to solve this problem, and finally translates the answer back into geometric terms. In the case of a line, one can model it as the set of real numbers.
To handle the plane, start with two perpendicular lines in the plane. Their point of intersection is denoted 0. The first line is called the x-axis and the second line is called the y-axis. Using the same unit distance as before, one can map the real numbers onto each of the two lines. With any pair (a, b) of real numbers, we can associate a point in the plane: From the point marked a on the x-axis, erect a line perpendicular to the x-axis. Similarly, from the point marked b on the y-axis, erect a line perpendicular to the y-axis. The point of intersection of these two lines is the point associated with (a, b). Assumption 1: The above mapping is a 1-1 and onto mapping between the set of all pairs (a,b) of real numbers and the points of the plane. Because of this assumption, we will usually not distinguish between the ordered pair (a, b) and its corresponding point in the plane. In particular, we will refer to the point as being the ordered pair (a,b). Remark: By using three pairwise perpendicular lines intersecting at a point 0, one can name points in three dimensional space as simply triples (a, b, c) of real numbers. Of course, one could then define n-dimensional space, as simply being the set of ordered n-tuples of numbers. Definition 1: The distance between the points P = (a,b) and Q = (c, d) is defined to be . This is called the distance formula. Although the formula is a bit complicated, this is the obvious definition as you can see by examining the diagram below in which PQ is simply the hypotenuse of a right triangle whose legs are of length |c - a| and |d - b| respectively. In light of the Pythagorean Theorem the formula for the length of this hypotenuse is the value given in the definition. Example 11: The points on either axis labeled a and b are of distance |b - a| from each other -- where distance is calculated by Definition 1. Example 12: The distance between the points (3, 5) and (2,7) is . Translation of Graphs If the point (x,y) satisfies the equation f(x, y) = 0, and if A and B are real numbers, then the point (x + A, y + B) satisfies the equation f(x - A, y - B) = 0 (because f((x+A) - A, (y+B) - B) = f(x, y) = 0). Another way of saying this is: if the graph of a function y = f(x) is moved A units to the right and B units to the right, then one obtains the graph of y - B = f(x - A). This amounts to the simple rules:
Graph Compression and Expansion If the point (x,y) satisfies the equation f(x, y) = 0, and if A and B are non-zero real numbers, then the point (Ax, By) satisifes the equation f(x/A, y/B) = 0 (because f((Ax)/A, (By)/B) = f(x, y) = 0). Another way of saying this is: if the graph of a function y = f(x) is expanded by a factor of A horizontally and by a factor of B vertically, then the result is the graph of y = Bf(x/A). Again, you can give simple rules:
Reflecting a graph across an axis If the point (x, y) satisfies the equation f(x, y) = 0, then the point (-x, y) obtained by reflecting the point (x, y) across the y-axis satisfies f(-x, y) = 0. Similarly, the point (x, -y) obtained by reflecting the point (x, y) across the x-axis satsifes the equation f(x, -y) = 0. 1.2.3 Lines, Circles, and ParabolasLines, circles, and other curves like parabolas should simply be the sets of solutions of certain algebraic equations. This appears to be the case. For example, we can define a vertical line to be the set of solutions of some equation x = a, where a is a constant. Similarly, a horizontal line is the set of solutions of an equation of the form for some constant b. Definition 2: A line is either a horizontal or vertical line or else it is the set of solutions of an equation of the form y = mx + b where m and b are constants. The equation y = mx + b (or x = a) is called the equation of the line which is the set of solutions of the equation. Note that this is a 1-1 correspondence between the set of lines and the set of equations of lines. In particular, different equations correspond to different lines; in fact, you can easily convince yourself that through any two points there is precisely one line. The constant m is called the slope of the line and the constant b is called the y-intercept of the line. Note that the y-intercept is the y-coordinate of the point where the line intersects the y-axis. The slope and y-intercept of a vertical line is not defined. There are many algebraically equivalent forms of the equation of a line; the particular one y = mx + b is called the slope-intercept form of the equation of the line. If for i = 1, 2 are two distinct points on the line y = mx + b, then one has for i = 1, 2 and, taking differences, one gets Solving for m gives: Proposition 6: The slope of the non-vertical line through and is given by . Example 13: The equation of the line through the two points and is given by This is the so-called 2 point form of the equation of the line. It is proved by substituting in the two points. Example 14: The equation of the line with slope m which passes through the point is . This is called the point-slope form of the equation of the line. Example 15: Consider the line through the two points (2, 3) and (5,7). Its slope is . Its equation in point-slope form is (or equivalently ). To find the y-intercept, just substitute x = 0 into either of these equations and solve for y; the y-intercept is 1/3. So the slope-intercept form of the equation of the line is y = (4/3)x + 1/3. To find points on this line, just substitute in arbitrarily chosen values of x and solve for the y-coordinate. Two lines which have no point of intersection are said to be parallel. Example 16: Two distinct lines are parallel if and only if they are either both vertical or else they both have the same slope. Suppose the two lines are not vertical. Let and be their equations and let's assume (x, y) is a point of intersection. Then it satisfies both equations. Equating them, we get and so . If the slopes were unequal, the coefficient of x would be non-zero and we could solve for and substitute this back into either of the original equations to obtain the value for y. If the slopes are not equal, it is easy to then verify that this pair is indeed a point of intersection, i.e. satisfies both equations. On the other hand, if the slopes were equal, then we would have which would mean that the lines were not distinct (because they have the same equation). In case the two lines are vertical, the equations are of the form and where . Clearly, no pair (x, y) could satisfy both of these equations; so distinct vertical lines are parallel. Finally, if one line has the equation and the other the equation , then the point (a, ma + b) is a point of intersection. This completes the proof of the assertion. We say that two lines are perpendicular if they intersect at right angles to each other. Example 17: Two lines are pendendicular if and only if one of the following is true:
It is easy to verify the result in case either of the two lines is vertical, either of the two lines is horizontal, or if the lines are are parallel or coincident. So, assume that none of these conditions are true. Then the two lines have slopes and their equations can be written for i = 1, 2 where is the point of intersection of the two lines. The lines have points A and B with x-coordinates, say, a + 1. Using the equations of the lines, we see that the points are and . By the Pythagorean Theorem, the lines are perpendicular if and only if . Using the distance formula, this amounts to If you multiply out and simplify this expression, you will see that it is equivalent to . Remark: The last example shows both the power and danger of analytic geometry. There was no understanding involved, the result came from brute algebraic computation. A circle with center (a, b) and radius r is the set of points (x,y) whose distance from (a, b) is precisely r. By the distance formula, this circle is the set of solutions of
The set of solutions of an equation of the form where , b, and c are constants is called a parabola. The following properties are easy to verify:
Example 18: Find the equation of the line which passes through the two points of intersection of the circle centered at the origin of radius 1 and the circle centered at (1,1) with radius 1. The equations of the two circles are and . If (x, y) is a point of intersection, it must satisfy both equations. Expanding out the second equation, one gets . Subtracting the first equation and simplifying gives the equation x + y = 1. If we knew that there were two points of intersection, then they would both satisfy this equation and it is the equation of a line; so it must be the desired line. (In order to verify that there are indeed two points of intersection, finish solving the system of equations and verify that the two solutions satisfy the original equations.) 1.2.4 Conic SectionsLet L be a non-vertical line through the origin. By rotating the line L around the y-axis, one obtains a pair of infinite cones with vertex at the origin. By intersecting this pair of cones with various planes, one obtains curves called ellipses, parabolas, and hyperbolas as well as some degenerate cases such as a single point or a single line. These intersections were called sections and so ellipses, parabolas, and hyperbolas were referred to as conic sections, i.e. sections of a cone. Nowadays the importance of conic sections is not that they arise by intersecting a cone with a plane, but rather that they can be used to categorize all curves whose equations are polynomials of degree 2. So, lines are the curves represented by equations which are polynomials of degree 1, and conic sections are the curves represented by equations which are polynomials of degree 2. In this section, we will define ellipses, parabolas, and hyperbolas without reference to sections of cones, and obtain equations for each in some special cases. 1.2.4.1 ParabolasWe have already defined parabolas to be the graphs of functions of the form . Let us give a more traditional definition: Definition 3: Let F be a point and D be a line in the plane which does not contain F. Then the parabola with focus F and directrix D is the set of points P = (x, y) such that the distance from P to F is the same as the distance from P to D. (By the distance from P to the line D, we mean the shortest distance from P to any point of D, i.e. the length of the line segment obtained by dropping a line through P perpendicular to D.) Now, let's look at a special case in which the vertex F = (0, d) and the line D is y = -d. If P = (x,y) is on the parabola with focus F and directrix D, then the distance formula tells us that When you multiply this out and collect terms, one sees that this is just or . Given the parabola, , we see that a = 1/(4d) and so the focus is at (0, d) = (0, 1/(4a)) and the directrix is y = -d = -1/(4a). Where is the focus and directrix of ? 1.2.4.2 EllipsesOne could define ellipses and hyperbolas in a manner exactly analogous to Definition 3, viz. Definition 3a: Let F be a point and D be a line in the plane which does not contain F and e be a positive real number. Then the conic section with focus F, directrix D, and eccentricity e is the set of points P = (x, y) such that the distance from P to F is equal to e times the distance from P to D. If e = 1, the conic section is called a parabola. The conic section is called an ellipse if e < 1 and a hyperbola if e > 1. Such a definition leads to rather complicated formulas, and so, instead, we will define Definition 4: Let and be two not necessarily distinct points in the plane and let a be a positive real number. Then the ellipse with foci and and semimajor axis a is the set of points P = (x, y) such that the sum of the distances from P to the foci is exactly 2a. Clearly, if a is less than half the distance between the foci, then the ellipse with semimajor axis a is the empty set. Again, let us consider a special case: Let and where is a real number. Let a be a real number greater than c. Then if P = (x, y) is a point of the ellipse with these points as foci and with semimajor axis a, then we have by the distance formula: This is the equation of the ellipse, but it is better to simplify it a bit. To do so, subtract the second square root from both sides and then square both sides to get: Moving all the terms except for the square root from the right side to the left side and simplifying gives Dividing both sides by 4 and squaring again gives which can be re-arranged to get Finally, since a is larger than c and both are positive, we can let b be a positive number such that . Substituting this into our last formula and dividing both sides by gives us the standard form of the equation of the ellipse: The quantity e = c/a is called the eccentricity of the ellipse. The quantity b is called the semiminor axis of the ellipse. In particular, if the eccentricity of the ellipse is 0, then the two foci coincide, the semimajor and semiminor axes are equal, and the ellipse is a circle whose radius is precisely the semimajor axis. Of course, one can also have ellipses with a vertical semimajor axis. If we simply interchange the roles of x and y in the above calculation, the final formula is the same, except that quantities a and b are interchanged. You distinguish the two cases by looking to see which of the two is the larger. You can also translate the graph of the equation to obtain ellipses centered at points (a, b) instead of at the origin (0, 0). As usual, the formulas look the same except that x is replaced with x - a and y is replace by y - b. Expansion and compression either horizontally or vertically just expands or contracts the values of a and b. 1.2.4.3 HyperbolasIn analogy with Definition 4, we have Definition 5: Let and be two not necessarily distinct points in the plane and let a be a positive real number. Then the ellipse with foci and and semimajor axis a is the set of points P = (x, y) such that the difference of the distances from P to the foci is exactly 2a. (Note that we have to subtract the smaller of two distances from the larger one in order to get 2a.) Taking the foci (0, -c) and (0, c) as before, it is easy to use the distance formula to write down the equation of the hyperbola. You should go through steps just as with the ellipse. When you are done, you will see that the equation of the hyperbola reduces down to where b is a positive number such that . As before, we call the eccentricity of the hyperbola. Note that e > 1 for hyperbolas and less than 1 for ellipses. 1.2.5 TrigonometryAngle Measurement: Angles are measured in radians. Start with the unit circle centered at the origin (0, 0). Make one side of the angle the positive x-axis. The other side of the angle is another ray from the origin, say 0A in the diagram below. The magnitude of the radian measure of the angle is equal to the length of the arc of the circle swept out as the positive x-axis is rotated through the angle to the ray 0A. The radian measure is positive if the arc is swept out in a counterclockwise direction and is negative otherwise. If we take instead of the unit circle, the circle centered at the origin of radius r, then the arc swept out by the same action is of length where is the radian measure of the angle. Again referring to the diagram above, if A is the point on the unit circle corresponding to the second ray of the angle, then its coordinates are by definition the cosine and sine of the angle , i.e. . Since A lies on the unit circle, we have the identity: . Further, it is clear by the symmetry of the circle that we have:
If B is the point on the ray 0A where the ray intersects the circle of radius r: , then is equal to the ratio of the x-coordinate of B and the the hypotenuse r (because of similar triangles). This explains the common definition of the cosine as being the ratio of the adjacent side to the hypotenuse; when using this, be careful, that that the adjacent side may be either a positive or negative number. Similarly, the sine of the same angle is the quotient of the opposite side over the hypotenuse (where the opposite side might be negative). These definitions allow one to create tables of trigonometric functions for practical use in solving right triangles. For example, Ptolemy computed tables in half degree increments. The remaining trigonometric functions are: Example 19: Dividing the identity , by and using the above definitions, one obtains the identity: Similarly, by dividing by , one obtains the identity:
Example 20: Let be the angle opposite the side of length 3 in a right triangle whose sides are of length 3, 4, and 5. Then the trigonometric functions of are: The values of the trigonometric functions at a number of commonly occurring angles are given in the table below:
Rather than memorize the table, it is easier to keep in mind the figure above used to define the trig functions and reconstruct the two standard triangles shown below. There are many more important results from trigonometry. The most important are: Proposition 7:(Addition Formulas) For arbitrary angles and , one has:
and Proposition 8: Let be any triangle and a, b, c be the lengths of the sides opposite angles A, B, and C respectively. Then
You will have an opportunity to prove these results in the exercises. All contents © copyright 2001 K. K. Kubota. All rights reserved |