4.1 Winkel und Kreise

Example 1

1. \displaystyle 30^\circ = 30 \cdot 1^\circ = 30 \cdot \frac{\pi}{180}\ \mbox{ radians } = \frac{\pi}{6}\ \mbox{ radians }
2. \displaystyle \frac{\pi}{8}\ \mbox { radians } = \frac{\pi}{8} \cdot (1 \; \mbox{radians}\,) = \frac{\pi}{8} \cdot \frac{180^\circ}{\pi} = 22{,}5^\circ

Example 2

1. The angles \displaystyle -55^\circ and \displaystyle 665^\circ indicate the same direction because Vorlage:Fristående formel
2. The angles \displaystyle \frac{3\pi}{7} and \displaystyle -\frac{11\pi}{7} indicate the same direction because Vorlage:Fristående formel
3. The angles \displaystyle 36^\circ and \displaystyle 216^\circ do not specify the same direction, but opposite directions since Vorlage:Fristående formel

Example 3

Example 4

1. The distance between \displaystyle (1,2) and \displaystyle (3,1) is Vorlage:Fristående formel
2. The distance between \displaystyle (-1,0) and \displaystyle (-2,-5) is Vorlage:Fristående formel

Example 5

1. The way radians have been defined means that the arc length is the radius multiplied by the angle measured in radians, Vorlage:Fristående formel

2. Determine the area of the circle segment.
   
   The circle segments share of the entire circle is Vorlage:Fristående formel 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:Fristående formel

Example 6

Example 7

1. 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:Fristående formel Since the point satisfies the circle equation it lies on the circle. 4.1 - Figur - Ekvationen (x - 4)² + y² = 13
2. Determine the equation for the circle that has its center 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:Fristående formel The circle equation is therefore Vorlage:Fristående formel 4.1 - Figur - Ekvationen (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 circle equation in the form Vorlage:Fristående formel 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:Fristående formel (the underlined terms shows the terms involved).

Complete the square for the terms containing \displaystyle y Vorlage:Fristående formel

The left-hand side is equal to Vorlage:Fristående formel

and moving over the 4 to to the right-hand side we get the circle equation Vorlage:Fristående formel

We can interpret this that the centre is \displaystyle (1,-2) and the radius is \displaystyle \sqrt{4}= 2.