Where is the dot?
Where is the Dot? printable sheet
Watch the film below.
Imagine the dot starts at the point $(1,0)$ and turns anticlockwise.
Estimate the height of the dot above the horizontal axis after it has turned through $45^\circ$.
Estimate the angle that the dot needs to turn in order to be exactly $0.5$ units above the horizontal axis.
Show how you can use Pythagoras' Theorem to calculate the height of the dot above the horizontal axis after it has turned through $45^\circ$.
Again, without resorting to Trigonometry, calculate the height of the dot above the horizontal axis after it has turned through $30^\circ$ and $60^\circ$?
Are there any other angles for which you can calculate the height of the dot above the horizontal axis?
The radius of the circle is $1$ unit long.
You can use the radius as the hypotenuse of the right-angled triangles.
Pythagoras' theorem states that $x^2+x^2=1^2$, i.e. $x^2=\frac{1}{2}$ so $x=\frac{1}{\sqrt{2}}$.
The next part of the question asked for similar ways of calculating the height of the dot after it had turned through $30^{\circ}$ and $60^{\circ}$.
Consider the right angled triangle we obtain after turning through $30^{\circ}$. If we reflect this triangle in the horizontal axis we obtain an equalateral triangle with sides of length $1$ as shown.
This implies that the height of the dot must be $\frac{1}{2}$. We deduce that at $60^{\circ}$ we end up with the following triangle:
By Pythagoras, the height of the dot must satisfy the equation $x^2+(\frac{1}{2})^2=1^2$ which implies that $x=\sqrt{1-\frac{1}{4}}=\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}$.
By symmetry, now that we know the height of the dot for angles of $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$, we can state the height for angles which are a multiple of $30^{\circ}$, $45^{\circ}$ or $60^{\circ}$. See if you can list these angles and the corresponding heights of the dot.
This problem offers students an opportunity to apply Pythagoras' Theorem.
It can also be used as a starting point for trigonometry:
- what happens to the height of the dot during the first $90^{\circ}$ of turn?
- what happens to the height of the dot when it turns beyond $90^{\circ}$?
- what can you say about the horizontal displacement of the dot as it turns through a full circle?