Complex numbers were first employed when the general formula for solutions of cubic and quartic equations gave rise to the definition of the square root of negative numbers. This had bewildered many prominent mathematicians, even Euler.
Now, we coin the number i such that i^2=-1. Also, this naturally led to the invention of -i. Note that however, we do not say that i>0 or -i<0. It is senseless to talk about their "positiveness", as proven by the following fact: If a>0, then a^2>0*a=0. However, i^2=-1<0, a contradiction.
Right then, people began investigating the property of such numbers. These numbers behave in a similar manner to surds (i.e. square root of real positive numbers), in a sense that they can be added and multiplied in an algebraic manner. They are demonstrated below:
Let a, b, c, d be real numbers. Then, the following properties of complex numbers hold true:
1) (a+bi)+(c+di)=(a+c)+(b+d)i
2) (a+bi)(c+di)=(ac-bd)+(ad+bc)i
3) (a+bi)c=ac+bci
For division, we first rationalize the denominator, then carry out multiplication in the above manner. Refer to the page on Surds.
Problems:
Saturday, September 25, 2010
Friday, September 24, 2010
Other Trigonometric Functions and the Pythagorean Identity
Besides sin, cos and tan, we also define 3 other trigonometric functions, namely, cot, sec and csc. We explore their properties here:
cotx*tanx=1 for all x where the product is defined
secx*cosx=1
cscx*sinx=1
Now, we will go through another fundamental trigonometric property of these 6 functions, namely the Pythagorean Identities.
1) (sinA)^2+(cosA)^2=1 for all real angles A.
In the right-angled triangle ABC, let the lengths of AB, BC and CA be c, a and b respectively.
By right-angled trigonometry in triangle ABC,
a=c*sinA, b=c*cosA
By the Pythagoras Theorem, we know that a^2+b^2=c^2
By the substitution above, we have (c*sinA)^2+(ccosA)^2=c^2. Dividing throughout by c^2 gives (sinA)^2+(cosA)^2=1.
Of course, this only works for A between 0 to 180 degrees. To extend the definition, refer to the Addition and Subtraction Formulas of sine and cosine. Also, note that sin0=0, sin90=1 and sin180=0. Similarly, cos0=1, cos90=0 and cos180=1. For more details on how to find cos0, sin0, cos180 and sin180, refer to the page on limits and continuity.
To obtain the other 2 formulas, we divide the equation by (sinA)^2 and (cosA)^2 respectively. We then obtain 1+(cotA)^2=(cscA)^2 and (tanA)^2+1=(secA)^2.
cotx*tanx=1 for all x where the product is defined
secx*cosx=1
cscx*sinx=1
Now, we will go through another fundamental trigonometric property of these 6 functions, namely the Pythagorean Identities.
1) (sinA)^2+(cosA)^2=1 for all real angles A.
In the right-angled triangle ABC, let the lengths of AB, BC and CA be c, a and b respectively.
By right-angled trigonometry in triangle ABC,
a=c*sinA, b=c*cosA
By the Pythagoras Theorem, we know that a^2+b^2=c^2
By the substitution above, we have (c*sinA)^2+(ccosA)^2=c^2. Dividing throughout by c^2 gives (sinA)^2+(cosA)^2=1.
Of course, this only works for A between 0 to 180 degrees. To extend the definition, refer to the Addition and Subtraction Formulas of sine and cosine. Also, note that sin0=0, sin90=1 and sin180=0. Similarly, cos0=1, cos90=0 and cos180=1. For more details on how to find cos0, sin0, cos180 and sin180, refer to the page on limits and continuity.
To obtain the other 2 formulas, we divide the equation by (sinA)^2 and (cosA)^2 respectively. We then obtain 1+(cotA)^2=(cscA)^2 and (tanA)^2+1=(secA)^2.
Basics of Geometry
Geometry is the study of shapes. The quantities it investigates includes angles, lengths, areas, volumes etc. The aim of geometry is to investigate several relationships between these variables in different setups.
To define these quantities, we need to define several terms:
A point is an object with no dimensions. It is the smallest thing there is, but that is not what we are concerned with.
A line is a straight 1-dimensional object. It extends in 2 directions. A ray extends only in 1 direction, and a segment is one with finite length.A curve is a 1-dimensional object that may not be straight. A curve is closed if a point is 'directly connected' with 2 other points and contains no ends, and open if there are ends (or potential ends in the infinite case).
A plane is a flat 2-dimensional object. It can either be finite or infinite. A surface, however, may not be flat, and can be curved or wavy in any way possible. A surface is still 2 dimensional, which means it is infinitely thin. A surface is closed if there is no identifiable 'boundary' (points lie on one side of a boundary but none on the other) across the whole surface, and open otherwise.
A solid is any 3-dimensional object. A solid can have holes inside it or along its surface. A hollow solid is a closed surface.
Now, we are ready to define angle, lengths, areas and volumes:
An angle can be formed between 2 planes, 2 lines or a plane and a line. It is meaningless to discuss about angles between curves or surfaces in general. If 2 lines intersect at a point, then the angle is a measure of how much one line can be rotated about that point to form the other line. The cases for the other 2 are slightly complicated, and will be discussed later. A rotation back to the original position is defined to be 360 degrees.
The length of a segment is a measure of how far apart the 2 endpoints are. It can either be measured on a straight line or a curve (known as the arc length in more advanced mathematics.) In physics, there are many useful length units, but we would not deal with units here.
The area of a surface is a measure on how big the surface is. It is analagous to the amount of paint you need to paint the whole surface.
The volume of an object measures how much space it occupies. Referring to Archimedes' claim "Eureka!", an object's volume can be measured by how much water it displaces in a bathtub.
Geometry studies the above relationships between angles, lengths, areas and volumes. Let us proceed to study them more in detail!
To define these quantities, we need to define several terms:
A point is an object with no dimensions. It is the smallest thing there is, but that is not what we are concerned with.
A line is a straight 1-dimensional object. It extends in 2 directions. A ray extends only in 1 direction, and a segment is one with finite length.A curve is a 1-dimensional object that may not be straight. A curve is closed if a point is 'directly connected' with 2 other points and contains no ends, and open if there are ends (or potential ends in the infinite case).
A plane is a flat 2-dimensional object. It can either be finite or infinite. A surface, however, may not be flat, and can be curved or wavy in any way possible. A surface is still 2 dimensional, which means it is infinitely thin. A surface is closed if there is no identifiable 'boundary' (points lie on one side of a boundary but none on the other) across the whole surface, and open otherwise.
A solid is any 3-dimensional object. A solid can have holes inside it or along its surface. A hollow solid is a closed surface.
Now, we are ready to define angle, lengths, areas and volumes:
An angle can be formed between 2 planes, 2 lines or a plane and a line. It is meaningless to discuss about angles between curves or surfaces in general. If 2 lines intersect at a point, then the angle is a measure of how much one line can be rotated about that point to form the other line. The cases for the other 2 are slightly complicated, and will be discussed later. A rotation back to the original position is defined to be 360 degrees.
The length of a segment is a measure of how far apart the 2 endpoints are. It can either be measured on a straight line or a curve (known as the arc length in more advanced mathematics.) In physics, there are many useful length units, but we would not deal with units here.
The area of a surface is a measure on how big the surface is. It is analagous to the amount of paint you need to paint the whole surface.
The volume of an object measures how much space it occupies. Referring to Archimedes' claim "Eureka!", an object's volume can be measured by how much water it displaces in a bathtub.
Geometry studies the above relationships between angles, lengths, areas and volumes. Let us proceed to study them more in detail!
Sunday, August 1, 2010
Addition and Subtraction Formulas
Ever wondered how to find sin(x+y) and cos(x+y) in terms of cosx, sinx, cosy and siny? We will go through the derivation.
Before looking through the proofs below, try to construct your own.
Note: The idea is always to drop perpendiculars, as right angle trigonometry comes in very handy. In most of these trigonometric formulas, in fact, it is very important to consider right angles.
1) cos(x+y)=cosx*cosy-sinx*siny
Let us construct a triangle ABC. We let D be the foot of perpendicular from A on BC. Then, let x=angle BAD and y=angle CAD. Also, let the length of AB=c, CD=e, AC=b, BD=d, AD=h. Below shows the diagram:
By cosine rule, we know that (d+e)^2=b^2+c^2-2bccos(x+y)
Also, by right angle trigonometry,
Triangle BAD: d=c*sinx, h=c*cosx
Triangle CAD: e=b*siny, h=b*cosy
Thus, we have (csinx)^2+2bc*sinx*siny+(bsiny)^2=b^2+c^2-2bc*cos(x+y)
Simplifying, we get 2bc(sinx*siny+cos(x+y))=(c*cosx)^2+(b*cosy)^2=2h^2
Thus, we have sinx*siny+cos(x+y)=cosx*cosy, and cos(x+y)=cosx*cosy-sinx*siny
2) sin(x+y)=sinx*cosy+cosx*siny
Using the same triangle, we consider areas.
Area of triangle ABC=(bc/2)*sin(x+y)=(d+e)(h/2)
Then, since h=c*cosx=b*cosy, d=csinx and e=bsiny, we have
h^2*sin(x+y)=(c*sinx+b*siny)*h*cosx*cosy=h^2(sinx*cosy+cosx*siny)
Hence, sin(x+y)=sinx*cosy+cosx*siny
From those identities, we can also write cos(x-y) and sin(x-y) in terms of cosx, cosy, sinx and siny. Try to find the final expressions.
1) Find the value of sin15 without using a calculator.
2) What is the value of sin90 and sin0? cos90 and cos0?
3) Show that cos(-x)=cosx and sin(-x)=-sinx
4) Remember the definition of a function? Plot a graph of f(x)=sinx and g(x)=cosx on the same axes. How did you extend the domain of the sin and cos functions?
5) Express tan(x+y) in terms of tanx and tany. Do the same for tan(x-y). Also, plot a graph of f(x)=tanx.
Before looking through the proofs below, try to construct your own.
Note: The idea is always to drop perpendiculars, as right angle trigonometry comes in very handy. In most of these trigonometric formulas, in fact, it is very important to consider right angles.
1) cos(x+y)=cosx*cosy-sinx*siny
Let us construct a triangle ABC. We let D be the foot of perpendicular from A on BC. Then, let x=angle BAD and y=angle CAD. Also, let the length of AB=c, CD=e, AC=b, BD=d, AD=h. Below shows the diagram:
By cosine rule, we know that (d+e)^2=b^2+c^2-2bccos(x+y)
Also, by right angle trigonometry,
Triangle BAD: d=c*sinx, h=c*cosx
Triangle CAD: e=b*siny, h=b*cosy
Thus, we have (csinx)^2+2bc*sinx*siny+(bsiny)^2=b^2+c^2-2bc*cos(x+y)
Simplifying, we get 2bc(sinx*siny+cos(x+y))=(c*cosx)^2+(b*cosy)^2=2h^2
Thus, we have sinx*siny+cos(x+y)=cosx*cosy, and cos(x+y)=cosx*cosy-sinx*siny
2) sin(x+y)=sinx*cosy+cosx*siny
Using the same triangle, we consider areas.
Area of triangle ABC=(bc/2)*sin(x+y)=(d+e)(h/2)
Then, since h=c*cosx=b*cosy, d=csinx and e=bsiny, we have
h^2*sin(x+y)=(c*sinx+b*siny)*h*cosx*cosy=h^2(sinx*cosy+cosx*siny)
Hence, sin(x+y)=sinx*cosy+cosx*siny
From those identities, we can also write cos(x-y) and sin(x-y) in terms of cosx, cosy, sinx and siny. Try to find the final expressions.
Try the following exercises
1) Find the value of sin15 without using a calculator.
2) What is the value of sin90 and sin0? cos90 and cos0?
3) Show that cos(-x)=cosx and sin(-x)=-sinx
4) Remember the definition of a function? Plot a graph of f(x)=sinx and g(x)=cosx on the same axes. How did you extend the domain of the sin and cos functions?
5) Express tan(x+y) in terms of tanx and tany. Do the same for tan(x-y). Also, plot a graph of f(x)=tanx.
Sunday, July 25, 2010
Sine and Cosine Rules
As with all other mathematics, there has to be applications of trigonometry. Here are the 3 basic problems central to navigation, architecture, and physics:
1) If we knew the length of 1 side and any 2 angles of the triangle, could we find the last angle and the length of the other 2 sides?
2) If we knew the length of any 2 sides and the angle in between the two sides of the triangle, could we find the other two angles and the length of the other side?
3) If we knew the length of all 3 sides of the triangle, could we find all 3 angles?
Try to come up with a method to these problems. Is it possible to work out the values without actually drawing an accurate diagram?
Mathematicians have again worked out 2 important observations regarding the above problems. They are known as the sine rule and cosine rule.
Note: In the proofs below, we chose only 1 vertex to drop the perpendicular from. In fact, any of the 3 would do, and we generally use all 3 to get the complete observation. However, we would only go through the proof once, and the reasoning is the same for the other 2 vertices.
Sine Rule:
In the diagram above, we drop a perpendicular from C onto AB, and label that point D. We will find h in terms of A, a, B and b in 2 different ways:
As with the trigonometry of right angles,
Triangle BCD: sin(B)=h/a
Triangle ACD: sin(A)=h/b
Then, h=asin(B)=bsin(A). In conclusion, we have a/sin(A)=b/sin(B)=c/sin(C).
Cosine Rule:
In the diagram above, we drop the perpendicular from B onto AC, and label that point D. The lengths are defined as in the diagram.
By applying Pythagoras's Theorem,
Triangle ABD: h^2=c^2-n^2
Triangle BCD: h^2=a^2-(b-n)^2
Combining them together and eliminating h^2, we get c^2-a^2=n^2-(b-n)^2=n^2-(b^2-2bn+n^2)=2bn-b^2.
Then, applying right-angle trigonometry on triangle ABD, we get cos(A)=n/c, or n=ccos(A). Substituting this to the above gives us c^2-a^2=2bccosA-b^2, or a^2=b^2+c^2-2bccosA.
1) We label the lengths of the sides of the triangle opposite vertices A, B and C be a, b and c respectively, and the angles A, B and C.
a) Find the values of b, c and C if a=6, A=30 degrees and B=70 degrees.
b) Find the values of A, B and C if a=4, b=7 and c=8.
2) What is the minimum number of the variables A, B, C, a, b and c do we need to know the values of do we need to find the rest of them? Prove your conjecture.
3) Find the possible values of A, C and c given that a=5, b=7 and B=60 degrees. Why are there 2 solutions?
4) Let AD be an angle bisector of angle BAC in triangle ABC. D lies on BC. Show that AB*CD=AC*BD. This is known as the Angle Bisector Theorem.
5) In triangle ABC, the sides of the triangle have lengths x-1, x and x+1. If the largest angle is twice of the smaller angle, find the perimeter of the triangle.
1) If we knew the length of 1 side and any 2 angles of the triangle, could we find the last angle and the length of the other 2 sides?
2) If we knew the length of any 2 sides and the angle in between the two sides of the triangle, could we find the other two angles and the length of the other side?
3) If we knew the length of all 3 sides of the triangle, could we find all 3 angles?
Try to come up with a method to these problems. Is it possible to work out the values without actually drawing an accurate diagram?
Mathematicians have again worked out 2 important observations regarding the above problems. They are known as the sine rule and cosine rule.
Note: In the proofs below, we chose only 1 vertex to drop the perpendicular from. In fact, any of the 3 would do, and we generally use all 3 to get the complete observation. However, we would only go through the proof once, and the reasoning is the same for the other 2 vertices.
Sine Rule:
In the diagram above, we drop a perpendicular from C onto AB, and label that point D. We will find h in terms of A, a, B and b in 2 different ways:
As with the trigonometry of right angles,
Triangle BCD: sin(B)=h/a
Triangle ACD: sin(A)=h/b
Then, h=asin(B)=bsin(A). In conclusion, we have a/sin(A)=b/sin(B)=c/sin(C).
Cosine Rule:
In the diagram above, we drop the perpendicular from B onto AC, and label that point D. The lengths are defined as in the diagram.
By applying Pythagoras's Theorem,
Triangle ABD: h^2=c^2-n^2
Triangle BCD: h^2=a^2-(b-n)^2
Combining them together and eliminating h^2, we get c^2-a^2=n^2-(b-n)^2=n^2-(b^2-2bn+n^2)=2bn-b^2.
Then, applying right-angle trigonometry on triangle ABD, we get cos(A)=n/c, or n=ccos(A). Substituting this to the above gives us c^2-a^2=2bccosA-b^2, or a^2=b^2+c^2-2bccosA.
Now, try the problems that follow:
1) We label the lengths of the sides of the triangle opposite vertices A, B and C be a, b and c respectively, and the angles A, B and C.
a) Find the values of b, c and C if a=6, A=30 degrees and B=70 degrees.
b) Find the values of A, B and C if a=4, b=7 and c=8.
2) What is the minimum number of the variables A, B, C, a, b and c do we need to know the values of do we need to find the rest of them? Prove your conjecture.
3) Find the possible values of A, C and c given that a=5, b=7 and B=60 degrees. Why are there 2 solutions?
4) Let AD be an angle bisector of angle BAC in triangle ABC. D lies on BC. Show that AB*CD=AC*BD. This is known as the Angle Bisector Theorem.
5) In triangle ABC, the sides of the triangle have lengths x-1, x and x+1. If the largest angle is twice of the smaller angle, find the perimeter of the triangle.
Introduction to Trigonometry
Many mathematicians knew about similar triangles and angle measurements. However, they had faced a problem, as stated below:
If I know the 3 sides of a triangle, am I able to find all 3 angles?
In the past, the only thing people could do was to construct the triangle with a compass and measurements. They initially defined 3 quantities related to the angle measured in a right angle triangle, known as the sine, cosine and tangent:
As in the diagram above, we define sin(A)=a/h, cos(A)=b/h and tan(A)=a/b. Later, they came up with a table of values that would be convenient for reference later on. An image of it is displayed below:
We will discuss some of their properties in later tutorials.
If I know the 3 sides of a triangle, am I able to find all 3 angles?
In the past, the only thing people could do was to construct the triangle with a compass and measurements. They initially defined 3 quantities related to the angle measured in a right angle triangle, known as the sine, cosine and tangent:
As in the diagram above, we define sin(A)=a/h, cos(A)=b/h and tan(A)=a/b. Later, they came up with a table of values that would be convenient for reference later on. An image of it is displayed below:
We will discuss some of their properties in later tutorials.
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