The Unit Circle

Section 10.5 The Unit Circle

The trigonometric functions sine, cosine, and tangent each play a specific role in the geometry of the unit circle. The key concepts in this section are summarized in the video below.

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Spend a moment reviewing Definition 10.1.1. Each trigonometric function is a certain ratio of $x\text{,}$ $y\text{,}$ and $r\text{,}$ where $(x,y)$ is a point on the circle of radius $r \gt 0$ centered at the origin $(0,0)\text{.}$ In particular, $\sin(\theta) = y/r$ and $\cos(\theta) = x/r\text{.}$ Rewritting these as,

\begin{gather*} y = r\sin(\theta)\\ x = r\cos(\theta) \end{gather*}

we see that the sine and cosine determine the coordinates of points on the circle.

Moreover, $\tan(\theta) = y/x$ is the slope of the line formed by the terminal side.

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Example 10.5.1. Find a Point on a Circle.

Using a calculator, estimate the coordinates of the point $(x,y)$ on the circle of radius $4$ determined by the angle $207^\circ$ in standard position. Then find the slope of the dashed line formed by the terminal side of the angle.

Solution.

\begin{equation*} x = r \cos(207^\circ) \approx 4 times (-0.891) = -3.56 \end{equation*}

\begin{equation*} x = r \sin(207^\circ) \approx 4 times (-0.454) = -1.82 \end{equation*}

\begin{equation*} m = \tan(207^\circ) \approx 0.510 \end{equation*}

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If we restrict to the unit circle with $r=1\text{,}$ we obtain a very direct interpretation of the three trigonometric functions.

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Theorem 10.5.2. Unit Circle Trigonometry.

Let $\theta$ be an angle in standard position. Let $P(x,y)$ be the corresponding point on the unit circle $x^2+y^2 =1$ determined by the terminal ray of $\theta\text{.}$

Then

\begin{equation*} \cos(\theta) = \text{$x$-coordinate of $P$} \end{equation*}

\begin{equation*} \sin(\theta) = \text{$y$-coordinate of $P$} \end{equation*}

\begin{equation*} \tan(\theta) = \text{slope of terminal ray} \end{equation*}

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Example 10.5.3. Interpreting Sine, Cosine, and Tangent.

Sketch an angle of 2 (radians) and use a calculator to determine the coordinates of the point it determines on the unit circle.

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Solution.

Start with the unit circle. Measure an arc along the circle of 2 units in length--each blue segment is one unit long--to obtain an angle of 2 radians.

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Using my calculator in radian mode, we find

\begin{equation*} \cos(2) \approx -0.42 \end{equation*}

\begin{equation*} \sin(2) \approx 0.91 \end{equation*}

\begin{equation*} \tan(2) \approx -2.19 \end{equation*}

So that, the point $(x,y)$ is close to the point $(-0.42,0.91)$ and the slope of the dashed line formed by the terminal side of the angle is about $-2.19\text{.}$ Take note to confirm that these results have reasonable values and correct signs by estimating in the figure.

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