Draw Angles & Find Values in Unit Circle

• MHB
• mathmari
In summary, drawing angles and finding values in the unit circle involves understanding the relationship between angles and their corresponding trigonometric values. The unit circle is a circle with a radius of one, and the values of sine, cosine, and tangent for different angles can be found by using the coordinates of the points where the angle intersects the unit circle. This process allows for a visual representation of the trigonometric functions and is useful in solving various mathematical and geometric problems involving angles.
mathmari
Gold Member
MHB
Hey! :giggle:

Make a drawing for each of the values of the angle below indicating the angle at the unit circle (in other words: $\text{exp} (i \phi )$) and its sine, show cosine, tangent and cotangent.

Give these four values explicitly in every case (you are allowed to use elementary geometry and the Pythagorean theorem).

$$\phi=\frac{\pi}{6}, \ \ \phi=\frac{\pi}{4}, \ \ \phi=\frac{2\pi}{3}, \ \ \phi=\frac{5\pi}{6}, \ \ \phi=-\frac{2\pi}{3}, \ \ \phi=-\frac{\pi}{3}$$

So at a unit circle we draw an angle $\phi$ and then we get a drawing like the following, right? But what does it mean to give these four values explicitly? :unsure:

mathmari said:
So at a unit circle we draw an angle $\phi$ and then we get a drawing like the following, right? But what does it mean to give these four values explicitly? :unsure:
Hey mathmari!

Nice picture! (Sun)
I believe we're supposed to make a separate drawing for each of the angles, and give the corresponding values, such as $\sin\frac\pi 6=\frac 12$.

Klaas van Aarsen said:
I believe we're supposed to make a separate drawing for each of the angles, and give the corresponding values, such as $\sin\frac\pi 6=\frac 12$.

I have found an online interactive tool to show all the trigonometric functions : https://www.matheretter.de/rechner/trigonometrie

But this shows only one function at a time, do you maybe know if these is a similar tool that shows all trigonometric function in one picture?

:unsure:

mathmari said:
I have found an online interactive tool to show all the trigonometric functions : https://www.matheretter.de/rechner/trigonometrie

But this shows only one function at a time, do you maybe know if these is a similar tool that shows all trigonometric function in one picture?
Have you considered Desmos, Geogebra, or TikZ?

Klaas van Aarsen said:
Have you considered Desmos, Geogebra, or TikZ?

I tried now Desmos :

:unsure:

I found this TikZ example on stack exchange:
https://tex.stackexchange.com/quest...-and-tangent-to-calculate-coordinates-in-tikz
\begin{tikzpicture}
\usetikzlibrary{angles,quotes}
\tikzset{My Grid/.style={help lines,color=blue!50}}

\draw[My Grid] (-4,-4) grid (4,4);
\draw (-5,0) node[ left ] {$(-5,0)$} -- (5,0) node[ right ] {$(5,0)$};
\draw (0,-5) node[ below ] {$(0,-5)$} -- (0,5) node[ above ] {$(0,5)$};
\draw (0,0) circle [ radius=3cm ];

\coordinate(O)at(0,0);
\draw[red, very thick] (30:3cm)coordinate(A)
--({3*cos(30)},0)coordinate(B);

\draw [very thick,orange] (3,0) -- (3,{3*tan(30)})coordinate(C);
\pic[fill=green!50!black,
angle eccentricity=1.2,
"$$\alpha$$"] {angle=B--O--A};
\draw (O)--(C);
\end{tikzpicture}
Code:
\begin{tikzpicture}
\usetikzlibrary{angles,quotes}
\tikzset{My Grid/.style={help lines,color=blue!50}}

\draw[My Grid] (-4,-4) grid (4,4);
\draw (-5,0) node[ left ] {$(-5,0)$} -- (5,0) node[ right ] {$(5,0)$};
\draw (0,-5) node[ below ] {$(0,-5)$} -- (0,5) node[ above ] {$(0,5)$};
\draw (0,0)  circle [ radius=3cm ];

\coordinate(O)at(0,0);
\draw[red, very thick] (30:3cm)coordinate(A)
--({3*cos(30)},0)coordinate(B);

\draw [very thick,orange] (3,0) -- (3,{3*tan(30)})coordinate(C);
\pic[fill=green!50!black,
angle eccentricity=1.2,
"$$\alpha$$"] {angle=B--O--A};
\draw (O)--(C);
\end{tikzpicture}

We can edit it with the TikZ Live Editor:
https://tikzimages.mathhelpboards.com/tikz/tikzlive.html

Last edited:

1. What is the unit circle and how is it used to draw angles?

The unit circle is a circle with a radius of 1 unit, centered at the origin of a graph. It is used to visualize and understand angles in the Cartesian coordinate system. To draw angles in the unit circle, the angle is measured from the positive x-axis in a counterclockwise direction and the point where the angle intersects the unit circle is used to find the sine, cosine, and tangent values of the angle.

2. How do I find the values of sine, cosine, and tangent for a given angle in the unit circle?

To find the values of sine, cosine, and tangent for a given angle in the unit circle, you can use the SOH-CAH-TOA method. SOH stands for "sine equals opposite over hypotenuse", CAH stands for "cosine equals adjacent over hypotenuse", and TOA stands for "tangent equals opposite over adjacent". By using these ratios, you can find the values of sine, cosine, and tangent for any angle in the unit circle.

3. Can I use the unit circle to find values for angles greater than 360 degrees?

Yes, the unit circle can be used to find values for angles greater than 360 degrees. Since the unit circle is a continuous circle, angles greater than 360 degrees can be represented by multiple revolutions around the circle. For example, an angle of 540 degrees can be represented as 1 and a half revolutions around the unit circle, and the values for sine, cosine, and tangent can still be found using the same methods.

4. Can I use the unit circle to find values for negative angles?

Yes, the unit circle can be used to find values for negative angles. Negative angles can be represented by rotating clockwise around the unit circle, and the values for sine, cosine, and tangent can still be found using the same methods as for positive angles.

5. How does the unit circle help in solving trigonometric equations?

The unit circle helps in solving trigonometric equations by providing a visual representation of angles and their corresponding values for sine, cosine, and tangent. This can help in understanding and solving equations involving trigonometric functions, as well as in graphing these functions. Additionally, the unit circle can be used to find reference angles, which can simplify equations and make them easier to solve.

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