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Wednesday, October 21, 2009

2. The Straight Line

Also on this page:

Point-slope form
General Form

Slope-Intercept Form of a Straight Line

The slope-intercept form (otherwise known as "gradient, y-intercept" form) of a line is given by:

y = mx + b

This tells us the slope of the line is m and the y-intercept of the line is b.

Example:

The line y = 2x + 4 has

slope m = 2 and
y-intercept b = 4.

We do not need to set up a table of values to sketch this line. Starting at the y-intercept (y = 4), we sketch our line by going up 2 units for each unit we go to the right (since the slope is 2 in this example). To find the x-intercept, we let y = 0.

2x + 4 = 0

x = -2

We notice that this is a function. That is, each value of x that we have gives one corresponding value of y.

See more on Functions and Graphs.

Point-Slope Form of a Straight Line

We need other forms of the straight line as well. A useful form is the point-slope form (or point - gradient form). We use this form when we need to find the equation of a line passing through a point (x₁, y₁) with slope m:

y − y₁ = m(x − x₁)

Example:

Need Graph Paper?

Find the equation of the line that passes through (-2, 1) with slope of -3.

This LiveMath document lets us play with this idea. Let's do this first.

LIVEMath

Now for the normal answer:

Answer

General Form

Another form of the straight line which we come across is general form:

Ax + By + C = 0

It can be useful for drawing lines by finding the y-intercept (put x = 0) and the x-intercept (put y = 0).

We also use General Form when finding Perpendicular Distance from a Point to a Line.

Example:

Draw the line 2x + 3y + 12 = 0.

Answer

Exercises

1. What is the equation of the line perpendicular to the line joining (4, 2) and (3, -5) and passing through (4, 2)?

[Need a reminder? See the section on Slopes of Perpendicular Lines.]

Answer

2. If 4x − ky = 6 and 6x + 3y + 2 = 0 are perpendicular, what is the value of k?

Answer

1. Basic Definitions http://www.intmath.com/Plane-analytic-geometry/1_Basic-definitions.php

Distance Formula

Recall Pythagoras' Theorem:

Also on this page:

Gradient (slope)
Inclination
Parallel Lines
Perpendicular Lines

Need Graph Paper?

math expression

For a right-angled triangle with hypotenuse length c,

We use this to find the distance between any two points (x₁, y₁) and (x₂, y₂) on the cartesian plane:

The Cartesian Plane

The cartesian plane was named after Rene Descartes. It is also called the x-y plane.

See more about Descartes in Functions and Graphs.

The point (x₂, y₁) is at the right angle. We can see that:

The distance between the points (x₁, y₁) and (x₂, y₁) is simply x₂ − x₁ and
The distance between the points (x₂, y₂) and (x₂, y₁) is simply y₂ − y₁.

Using Pythagoras' Theorem we have the distance between (x₁, y₁) and (x₂, y₂) given by:

Example 1:

Find the distance between the points (3, -4) and (5, 7).

First, let's see this in LiveMath.

LIVEMath

Now for the normal answer:

Answer

Example 2:

Find the distance between the points (3, -1) and (-2, 5).

Answer

Gradient (or slope)

The gradient of a line is defined as

In this triangle, the gradient of the line is given by:

In general, for the line joining the points (x₁, y₁) and (x₂, y₂):

math expression

We see from the diagram above, that the gradient (usually written m) is given by:

Example:

Find the slope of the line joining the points (-4, -1) and (2, -5).

Answer

Positive and Negative Slopes

In general, a positive slope indicates the value of the dependent variable increases as we go left to right:

math expression

[The dependent variable in the above graph is the y-value.]

A negative slope means that the value of the dependent variable is decreasing as we go left to right:

math expression

Here is an animation of this using LiveMath.

LIVEMath

Inclination

We have a line with slope m and the angle that the line makes with the x-axis is α.

From trigonometry, we recall that the tan of angle α is given by:

Now, since slope is also defined as opposite/adjacent, we have:

inclination

This gives us the result:

tan α = m

Then we can find angle α using

α = arctan m

(That is, α = tan^-1m)

This angle α is called the inclination of the line.

Example 1:

Find the inclination of the line with slope 2.

Answer

NOTE: The size of angle α is (by definition) only between 0° and 180°.

Example 2:

Find the slope of the line with inclination α = 137°.

Answer

Let's see Gradient and Inclination using LiveMath.

LIVEMath

Parallel Lines

math expression

Lines which have the same slope are parallel.

If a line has slope m₁ and another line has slope m₂ then the lines are parallel if

m₁ = m₂

Here is a LiveMath animation showing that if the gradient stays the same and we only change the y-intercept, the lines are parallel.

LIVEMath

Perpendicular Lines

If a line has slope m₁ and another line has slope m₂ then the lines are perpendicular if

m₁ × m₂= -1

In the example at right, the slopes of the lines are 2 and -0.5 and we have:

2 × -0.5 = -1

So the lines are perpendicular.

Let's see a LiveMath example:

LIVEMath

Example:

A line l has slope m = 4.

a) What is the slope of a line parallel to l?

b) What is the slope of a line perpendicular to l?

Answers

Special Cases

What if one of the lines is parallel to the y-axis?

For example, the line y = 3 is parallel to the x-axis and has slope 0. The line x = 3.6 is parallel to the y-axis and has an undefined slope.

The lines are clearly perpendicular, but we cannot find the product of their slopes. In such a case, we cannot draw a conclusion from the product of the slopes, but we can see immediately from the graph that the lines are perpendicular.

The same situation occurs with the x- and y-axes. They are perpendicular, but we cannot calculate the product of the 2 slopes, since the slope of the y-axis is undefined.

Exercises

What is the distance between (-1, 3) and (-8, -4)?
A line passes through (-3, 9) and (4, 4). Another line passes through (9, -1) and (4, -8). Are the lines parallel or perpendicular?
Find k if the distance between (k,0) and (0, 2k) is 10 units.

Answers

How many trig functions are there?

How many basic trigonometric functions are there?

The calculator answer: 3

A typical calculator has three trig functions if it has any: sine, cosine, and tangent. The other three that you may see — cosecant, secant, and cotangent — are the reciprocals of sine, cosine, and tangent respectively. Calculator designers expect you to push the cosine key followed by the reciprocal key if you want a secant, for example.

Tuesday, October 20, 2009

The Trigonometry of the Triangle

http://www.codecogs.com/reference/maths/trigonometery/the_trigonometry_of_the_triangle.php
There are a number of equations associated with triangles. Of these, the best known are the Sine and Cos formulae.

The Sine Formula.

Consider the Triangle ABC with its Circumcircle. Draw the diameter BX through B

Angle BAX = 90 degrees and angle AXB = angle ACB

From the diagram it can be seen that c = 2R sin C

(1)

$\displaystyle \therefore\;\;\;\;\;\frac{c}{sin\,C}\;=\;2R$

Therefore by symmetry:-

(2)

$\displaystyle \mathbf{\frac{a}{sin\,A}\;=\;\frac{b}{sin\,B}\;=\;\frac{c}{sin\,C}\;=\;2R}$

Obtuse Case If A is obtuse, angle BXC = 180 - A

(3)

$\displaystyle a\;=\;2R\,sin\,(180^0\;-\;A)\;=\;2R\,sin\,A$

(4)

$\displaystyle \mathbf{\frac{a}{sin\,A}\;=\;\frac{b}{sin\,B}\;=\;\frac{c}{sin\,C}\;=\;2R}$

The Cosine Formula

ABC is an acute-angled triangle of height h

(5)

$\displaystyle Then\;\;\;\;\;\;\;\;CD\;=\;b\;cos\,C$

(6)

$\displaystyle \therefore\;\;\;\;\;\;BD\;=\;a\;-\;b\;cos\,C$

Using Pythagoras:-

(7)

$\displaystyle h^2\;=\;b^2\;-\;(b\;cos\,C)^2$

(8)

$\displaystyle and\;\;\;\;\;\;h^2\;=\;c^2\;-\;(a\;-\;b\;cos\,C)^2$

(9)

$\displaystyle \therefore\;\;\;\;\;b^2\;-\;b^2\,cos^2\,C\;=\;c^2\;-\;a^2\;-\;b^2\;cos^2\,C\;+\;2ab\;cos\,C$

(10)

$\displaystyle or\;\;\;\;\;\;\;\mathbf{c^2\;=\;a^2\;+\;b^2\;-\;2ab\;cos\,C}$

NOTE This equation can be re-written in terms of either angle A or B If C is obtuse

(11)

$\displaystyle CD\;=\;b\;cos\,(180^0\;-\;C)$

(12)

$\displaystyle BD\;=\;a\;+\;b\;cos\,(180^0\;-\;C)$

(13)

$\displaystyle \therefore\;\;\;\;\;h^2\;=\;b^2\;-\;b^2\;cos^2\,(180^0\;-\;C)$

(14)

$\displaystyle and\;\;\;\;\;h^2\;=\;c^2\;-\;\left(a\;+\;b\;cos\,[180^0\;-\;C] \right)^2$

(15)

$\displaystyle \therefore\;\;\;\;b^2\;-\;b^2\;cos^2(180^0\;-\;C)\;= c^2\;-\;a^2\;-\;b^2\;cos^2\,(180^0\;-\;C)\;-\;2ab\;cos\,(180^0\;-\;C)$

(16)

$\displaystyle or\;\;\;\;\;c^2\;=\;a^2\;+\;b^2\;+\;2ab\;cos\,(180^0\;-\;C)$

(17)

$\displaystyle Thus\;\;\;\;\;\mathbf{c^2\;=\;a^2\;+\;b^2\;-\;2ab\;cos\,C}$

It should be noted that the same equation can be applied in both cases.

Area Of A Triangle

The area of a Triangle is a half base times height.

(48)

$\displaystyle i.e.\;\;\;\;\;\;Area\;=\;\frac{1}{2}\;BC\times AD$

(49)

$\displaystyle But\;\;\;\;\;AD\;=\;b\;sin\;C$

(50)

$\displaystyle \therefore\;\;\;\;\;\;\;\mathbf{Area\;=\;\frac{1}{2}\,ab\,sinC\;=\;\frac{1}{2}\,bc\,sinA\;=\;\frac{1}{2}\,ac\,sin\,B}$

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Wednesday, October 21, 2009

2. The Straight Line

Slope-Intercept Form of a Straight Line

Example:

Point-Slope Form of a Straight Line

Example:

Need Graph Paper?

General Form

Example:

Exercises

1. Basic Definitions http://www.intmath.com/Plane-analytic-geometry/1_Basic-definitions.php

Distance Formula

Also on this page:

Need Graph Paper?

The Cartesian Plane

Example 1:

Example 2:

Gradient (or slope)

Example:

Positive and Negative Slopes

Inclination

Example 1:

Example 2:

Parallel Lines

Perpendicular Lines

Example:

Special Cases

Exercises

How many trig functions are there?

The calculator answer: 3

Tuesday, October 20, 2009

The Trigonometry of the Triangle

The Sine Formula.

The Cosine Formula

Area Of A Triangle

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