M.1.A
Video
It’s about the angular position. Consider the position of the second arm on a clock face. Proceed as follows.
Steps
1. Angular Position
Angle describes angular position
Conventions for angle \(\varphi\):
Domain: \(0^\circ \le \varphi < 360^\circ\)
Origin: \(\varphi=0^\circ\) at 3 o’clock position
Counting direction: Counter clockwise
Fill in the table below:
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
|---|---|
2 o’clock |
\(30^\circ\) |
12 o’clock |
\(90^\circ\) |
6 o’clock |
\(\ldots\) |
11 o’clock |
\(\ldots\) |
4 o’clock |
\(\ldots\) |
5 o’clock |
\(\ldots\) |
Solution
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
|---|---|
2 o’clock |
\(30^\circ\) |
12 o’clock |
\(90^\circ\) |
6 o’clock |
\(270^\circ\) |
11 o’clock |
\(120^\circ\) |
4 o’clock |
\(330^\circ\) |
5 o’clock |
\(300^\circ\) |
2. Periodicity
360 Degrees Periodicity
One arm position can be described with multiple angles. All angles below describe the “2 o’clock” arm position:
Fill in the table below.
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
\(360^\circ \le \varphi < 720^\circ\) |
|---|---|---|
2 o’clock |
\(30^\circ\) |
\(390^\circ\) |
12 o’clock |
\(90^\circ\) |
\(450^\circ\) |
6 o’clock |
\(\ldots\) |
\(\ldots\) |
11 o’clock |
\(\ldots\) |
\(\ldots\) |
4 o’clock |
\(\ldots\) |
\(\ldots\) |
5 o’clock |
\(\ldots\) |
\(\ldots\) |
Solution
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
\(360^\circ \le \varphi < 720^\circ\) |
|---|---|---|
2 o’clock |
\(30^\circ\) |
\(390^\circ\) |
12 o’clock |
\(90^\circ\) |
\(450^\circ\) |
6 o’clock |
\(270^\circ\) |
\(630^\circ\) |
11 o’clock |
\(120^\circ\) |
\(480^\circ\) |
4 o’clock |
\(330^\circ\) |
\(690^\circ\) |
5 o’clock |
\(300^\circ\) |
\(660^\circ\) |
3. Counting Direction
Negative Angles
One arm position can be described with multiple angles. All angles below describe the “2 o’clock” arm position:
The green arrow indicates the direction, in which \(\varphi\) grows.
The red arrow indicates the direction, in which \(-\varphi\) grows.
Fill in the table below.
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
\(0^\circ \le - \varphi < 360^\circ\) |
\(-360^\circ < \varphi \le 0^\circ\) |
|---|---|---|---|
2 o’clock |
\(30^\circ\) |
\(330^\circ\) |
\(-330^\circ\) |
12 o’clock |
\(90^\circ\) |
\(270^\circ\) |
\(-270^\circ\) |
6 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
11 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
4 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
5 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
Solution
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
\(0^\circ \le - \varphi < 360^\circ\) |
\(-360^\circ < \varphi \le 0^\circ\) |
|---|---|---|---|
2 o’clock |
\(30^\circ\) |
\(330^\circ\) |
\(-330^\circ\) |
12 o’clock |
\(90^\circ\) |
\(270^\circ\) |
\(-270^\circ\) |
6 o’clock |
\(270^\circ\) |
\(90^\circ\) |
\(-90^\circ\) |
11 o’clock |
\(120^\circ\) |
\(240^\circ\) |
\(-240^\circ\) |
4 o’clock |
\(330^\circ\) |
\(30^\circ\) |
\(-30^\circ\) |
5 o’clock |
\(300^\circ\) |
\(60^\circ\) |
\(-60^\circ\) |
4. Degrees and Radians
Fill in the table below.
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
\(0 \le \varphi < 2 \pi\) |
\(- 2 \pi < \varphi \le 0\) |
|---|---|---|---|
2 o’clock |
\(30 ^\circ\) |
\(\tfrac{1}{12} \cdot 2\pi\) |
\(-\tfrac{11}{12} \cdot 2\pi\) |
12 o’clock |
\(90 ^\circ\) |
\(\tfrac{3}{12} \cdot 2\pi\) |
\(-\tfrac{9}{12} \cdot 2\pi\) |
6 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
11 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
4 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
5 o’clock |
\(\ldots\) |
\(\ldots\) |
\(\ldots\) |
Solution
Ang. Position |
\(0^\circ \le \varphi < 360^\circ\) |
\(0 \le \varphi < 2 \pi\) |
\(- 2 \pi < \varphi \le 0\) |
|---|---|---|---|
2 o’clock |
\(30 ^\circ\) |
\(\tfrac{1}{12} \cdot 2\pi\) |
\(-\tfrac{11}{12} \cdot 2\pi\) |
12 o’clock |
\(90 ^\circ\) |
\(\tfrac{3}{12} \cdot 2\pi\) |
\(-\tfrac{9}{12} \cdot 2\pi\) |
6 o’clock |
\(270^\circ\) |
\(\tfrac{9}{12} \cdot 2\pi\) |
\(-\tfrac{3}{12} \cdot 2\pi\) |
11 o’clock |
\(120^\circ\) |
\(\tfrac{4}{12} \cdot 2\pi\) |
\(-\tfrac{8}{12} \cdot 2\pi\) |
4 o’clock |
\(330^\circ\) |
\(\tfrac{11}{12}\cdot 2\pi\) |
\(-\tfrac{1}{12} \cdot 2\pi\) |
5 o’clock |
\(300^\circ\) |
\(\tfrac{10}{12}\cdot 2\pi\) |
\(-\tfrac{2}{12} \cdot 2\pi\) |
5. Choices
Choose symbol, origin and counting direction
Winkel werden zur Beschreibung der Winkelposition des Zeigers verwendet. Hierbei wählt man die Bezeichnung des Winkels, den Nullpunkt und die Zählrichtung:
Fill in the table below.
Ang. Position |
\(0^\circ \le \psi < 360^\circ\) |
\(0 \le \psi < 2 \pi\) |
|---|---|---|
7 o’clock |
\(30 ^\circ\) |
\(\tfrac{1}{12} \cdot 2\pi\) |
9 o’clock |
\(90 ^\circ\) |
\(\tfrac{3}{12} \cdot 2\pi\) |
6 o’clock |
\(\ldots\) |
\(\ldots\) |
11 o’clock |
\(\ldots\) |
\(\ldots\) |
4 o’clock |
\(\ldots\) |
\(\ldots\) |
5 o’clock |
\(\ldots\) |
\(\ldots\) |
Solution
Ang. Position |
\(0^\circ \le \psi < 360^\circ\) |
\(0 \le \psi < 2 \pi\) |
|---|---|---|
7 o’clock |
\(30 ^\circ\) |
\(\tfrac{1}{12} \cdot 2\pi\) |
9 o’clock |
\(90 ^\circ\) |
\(\tfrac{3}{12} \cdot 2\pi\) |
6 o’clock |
\(0^\circ\) |
\(\tfrac{0}{12} \cdot 2\pi\) |
11 o’clock |
\(150^\circ\) |
\(\tfrac{5}{12} \cdot 2\pi\) |
4 o’clock |
\(300^\circ\) |
\(\tfrac{10}{12}\cdot 2\pi\) |
5 o’clock |
\(330^\circ\) |
\(\tfrac{11}{12}\cdot 2\pi\) |
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