4.1 Winkel und Kreise
Aus Online Mathematik Brückenkurs 1
Contents:
- Various angle measures (degrees, radians and revolutions)
- The Pythagorean theorem
- Formula for distance in the plane
- Equation of a circle
Learning outcomes:
After this section, you will have learned :
- To convert between degrees, radians and revolutions.
- To calculate the area and circumference of sectors of a circle.
- The concepts of right-angled triangles including its legs and hypotenuse.
- To formulate and use the Pythagorean theorem.
- To calculate the distance between two points in the plane.
- To sketch circles by completing the square in their equations.
- The concepts of unit circle, tangent, radius, diameter, circumference, chord and arc.
- To solve geometric problems that contain circles.
Angle measures
There are several different units for measuring angles, which are used in different contexts. The two most common within mathematics are degrees and radians.
- Degrees. If a complete revolution is divided into 360 parts, then each part is called 1 degree. Degrees are designated by \displaystyle {}^\circ.
- Radians. Another way to measure an angle, is to use the length of the arc which subtends the angle in relation to the radius as a measure of the angle. This unit is called radian. A revolution is \displaystyle 2\pi radians as the circumference of a circle is \displaystyle 2\pi r, where \displaystyle r is the radius of the circle.
A complete revolution is \displaystyle 360^\circ or \displaystyle 2\pi radians which means
Vorlage:Displayed math
These conversion relations can be used to convert between degrees and radians.
Example 1
- \displaystyle 30^\circ = 30 \cdot 1^\circ = 30 \cdot \frac{\pi}{180}\ \mbox{ radians } = \frac{\pi}{6}\ \mbox{ radians }
- \displaystyle \frac{\pi}{8}\ \mbox { radians } = \frac{\pi}{8} \cdot (1 \; \mbox{radians}\,) = \frac{\pi}{8} \cdot \frac{180^\circ}{\pi} = 22{,}5^\circ
In some contexts, it may be useful to talk about negative angles and angles greater than 360°. This means that the same direction can be designated by different angles that differ from each other by an integral number of revolutions.
Example 2
- The angles \displaystyle -55^\circ and \displaystyle 665^\circ indicate the same direction because Vorlage:Displayed math
- The angles \displaystyle \frac{3\pi}{7} and \displaystyle -\frac{11\pi}{7} indicate the same direction because Vorlage:Displayed math
- The angles \displaystyle 36^\circ and \displaystyle 216^\circ do not specify the same direction, but opposite directions since Vorlage:Displayed math
Formula for distance in the plane
The theorem of Pythagoras is one of the most famous theorems in mathematics and says that in a right-angled triangle with the legs \displaystyle a and \displaystyle b, and the hypotenuse \displaystyle c then
| The Pythagorean theorem: | 4.1 - Figure - The Pythagorean theorem |
Example 3
| The triangle on the right is
Vorlage:Displayed math and therefore hypotenuse \displaystyle c is equal to Vorlage:Displayed math | 4.1 - Figure - A right-angled triangle with sides 3, 4 and 5 |
The Pythagorean theorem can be used to calculate the distance between two points in a coordinate system.
Formula for distance:
The distance \displaystyle d between two points with coordinates \displaystyle (x,y) and \displaystyle (a,b) is Vorlage:Displayed math
The line joining the points is the hypotenuse of a triangle whose legs are parallel to the coordinate axes.
The legs of the triangle have lengths equal to the difference in the x- and y-directions of the points, that is \displaystyle |x-a| and \displaystyle |y-b|. The Pythagorean theorem then gives the formula for the distance.
Example 4
- The distance between \displaystyle (1,2) and \displaystyle (3,1) is Vorlage:Displayed math
- The distance between \displaystyle (-1,0) and \displaystyle (-2,-5) is Vorlage:Displayed math
Circles
A circle consists of all the points that are at a given fixed distance \displaystyle r from a point \displaystyle (a,b).
The distance \displaystyle r is called the circle´s radius and the point \displaystyle (a,b) is its centre. The figure below shows the other important concepts.
| 4.1 - Figur - Diameter | 4.1 - Figur - Tangent | 4.1 - Figure - The chord of a circle | 4.1 - Figur - Sekant | |||
| Diameter | Tangent | Chord | Secant | |||
| 4.1 - Figure - Arc of a circle | 4.1 - Figure - Circumference | 4.1 - Figure - Sector of a circle | 4.1 - Figure - Segment of a circle | |||
| Arc of a circle | circumference | sector of a circle | segment of a circle |
Example 5
A sector of a circle is given in the figure on the right.
|
- The way radians have been defined means that the arc length is the radius multiplied by the angle measured in radians, Vorlage:Displayed math
- Determine the area of the circle segment.
The circle segment´s share of the entire circle is Vorlage:Displayed math and this means that its area is \displaystyle \frac{5}{36} parts of the circle area, which is \displaystyle \pi r^2 = \pi 3^2 = 9\pi, i.e. Vorlage:Displayed math
A point \displaystyle (x,y) lies on the circle that has its centre at \displaystyle (a,b) and radius \displaystyle r, if its distance from the centre is equal to \displaystyle r. This condition can be formulated with the distance formula as
| Circle equation: | 4.1 - Figure - Equation of a circle |
Example 6
| 4.1 - Figure - The equation (x - 1)² + (y - 2)² = 9 |
| 4.1 - Figur - Ekvationen x² + (y - 1)² = 1 |
| 4.1 - Figure - The equation (x + 1)² + (y - 3)² = 5 |
Example 7
- Does the point \displaystyle (1,2) lie on the circle \displaystyle (x-4)^2 +y^2=13?
Inserting the coordinates of the point \displaystyle x=1 and \displaystyle y=2 in the circle equation, we have that Vorlage:Displayed math Since the point satisfies the circle equation it lies on the circle.4.1 - Figur - Ekvationen (x - 4)² + y² = 13 - Determine the equation for the circle that has its centre at \displaystyle (3,4) and goes through the point \displaystyle (1,0).
Since the point \displaystyle (1,0) lies on the circle, the radius of the circle must be equal to the distance of the point from \displaystyle (1,0) to the centre \displaystyle (3,4). The distance formula gives that this distance is Vorlage:Displayed math The circle equation is therefore Vorlage:Displayed math4.1 - Figure - The equation (x - 3)² + (y - 4)² = 20
Example 8
Determine the centre and radius of the circle with equation \displaystyle \ x^2 + y^2 – 2x + 4y + 1 = 0.
Let us try to write the equation of the circle in the form
Vorlage:Displayed math
because then we can directly read from this that the midpoint is \displaystyle (a,b) and the radius is \displaystyle r.
Start by completing the square for the terms containing \displaystyle x on the left-hand side Vorlage:Displayed math (the underlined terms shows the terms involved).
Complete the square for the terms containing \displaystyle y Vorlage:Displayed math
The left-hand side is equal to Vorlage:Displayed math
and moving over the 4 to to the right-hand side we get the equation Vorlage:Displayed math
We can interpret this as follows: The centre is at \displaystyle (1,-2) and the radius is \displaystyle \sqrt{4}= 2.
Study advice
The basic and final tests
After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
Keep in mind that:
Reviews
For those of you who want to deepen your studies or need more detailed explanations consider the following references:
Learn more about Pythagoras theorem in English Wikipedia
Read more in Mathworld about the circle
Useful web sites
Interactive experiments: the sine and cosine on the unit circle (Flash)
