5.1 The Unit Circle

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5.1.1 Learning Objectives

By the end of this section, you should be able to:

• verify that a point lies on the unit circle;
• locate terminal points for real numbers;
• find reference numbers and coterminal values;
• determine exact terminal points for common inputs; and
• apply symmetry and periodicity on the unit circle.

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5.1.2 Terminal Points on the Unit Circle

The unit circle is \(x^2+y^2=1\). Starting at \((1,0)\), travel a directed distance \(t\) around the circle. Counterclockwise travel is positive and clockwise travel is negative. The endpoint \(P(t)=(x,y)\) is the terminal point determined by \(t\). Because the radius is \(1\), the distance \(t\) is also the angle measure in radians.

Example 5.1 Checking a Point

Does \((3/5,4/5)\) lie on the unit circle?

Solution

\[ \left(\frac35\right)^2+\left(\frac45\right)^2=\frac9{25}+\frac{16}{25}=1. \]

\[ \boxed{\left(\frac35,\frac45\right)\text{ lies on the unit circle}} \]

Exercise 5.1 Does \((1/2,\sqrt3/2)\) lie on the unit circle?

Show answer

Answer: Yes, because \(1/4+3/4=1\).

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5.1.3 Common Terminal Points

Important terminal points come from \(30^\circ\), \(45^\circ\), and \(60^\circ\) reference triangles. Since a complete revolution has length \(2\pi\), the axis points occur at multiples of \(\pi/2\).

\(t\)	Terminal point
\(0\)	\((1,0)\)
\(\pi/6\)	\((\sqrt3/2,1/2)\)
\(\pi/4\)	\((\sqrt2/2,\sqrt2/2)\)
\(\pi/3\)	\((1/2,\sqrt3/2)\)
\(\pi/2\)	\((0,1)\)
\(\pi\)	\((-1,0)\)
\(3\pi/2\)	\((0,-1)\)
\(2\pi\)	\((1,0)\)

Example 5.2 Finding an Exact Terminal Point

Find the terminal point for \(t=5\pi/6\).

Solution

The reference number is \(\pi/6\). In Quadrant II, \(x<0\) and \(y>0\):

\[ \boxed{P(5\pi/6)=\left(-\frac{\sqrt3}{2},\frac12\right)}. \]

Exercise 5.2 Find \(P(7\pi/4)\).

Show answer

Answer: \(\boxed{(\sqrt2/2,-\sqrt2/2)}\)

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5.1.4 Coterminal Values and Periodicity

Inputs that differ by a whole number of revolutions have the same terminal point:

\[ P(t+2\pi k)=P(t),\qquad k\in\mathbb Z. \]

Such inputs are coterminal. Reduce a large positive or negative input by adding or subtracting multiples of \(2\pi\).

Example 5.3 Reducing an Input

Find the terminal point for \(t=17\pi/6\).

Solution

\[ \frac{17\pi}{6}-2\pi=\frac{5\pi}{6}. \]

Therefore,

\[ \boxed{P(17\pi/6)=\left(-\frac{\sqrt3}{2},\frac12\right)}. \]

Exercise 5.3 Find a coterminal value in \([0,2\pi)\) for \(-5\pi/3\).

Show answer

Answer: \(\boxed{\pi/3}\)

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5.1.5 Reference Numbers and Symmetry

The reference number is the shortest positive distance along the unit circle from the terminal point to the \(x\)-axis. It lies in \([0,\pi/2]\). The reference number determines the coordinate magnitudes; the quadrant determines their signs.

Example 5.4 Using a Reference Number

Find the reference number and terminal point for \(t=4\pi/3\).

Solution

The input lies in Quadrant III:

\[ \frac{4\pi}{3}-\pi=\frac{\pi}{3}. \]

Both coordinates are negative in Quadrant III:

\[ \boxed{\text{Reference number }\pi/3;\quad P(4\pi/3)=\left(-\frac12,-\frac{\sqrt3}{2}\right)}. \]

Exercise 5.4 Find the reference number for \(t=11\pi/6\).

Show answer

Answer: \(\boxed{\pi/6}\)

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5.1.6 Conceptual Takeaways

• Real numbers represent directed distances around the unit circle.
• A full revolution has length \(2\pi\).
• Coterminal inputs share a terminal point.
• Reference numbers determine coordinate magnitudes.
• Quadrant signs and symmetry extend first-quadrant values around the circle.

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5.1.7 Skills You Should Be Able to Do

• Verify unit-circle points.
• Locate exact terminal points.
• Reduce inputs using periodicity.
• Find reference numbers.
• Apply quadrant signs and symmetry.

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5.1.8 Practice Problems with Solutions

1. Verify that \((-5/13,12/13)\) lies on the unit circle.

   Show Solution

   \[ \left(-\frac5{13}\right)^2+\left(\frac{12}{13}\right)^2 =\frac{25+144}{169}=1. \]

   Therefore, \(\boxed{\text{the point lies on the unit circle}}\).

2. Find \(P(\pi/3)\).

   Show Solution

   From the common unit-circle points,

   \[ \boxed{P(\pi/3)=(1/2,\sqrt3/2)}. \]

3. Find \(P(3\pi/4)\).

   Show Solution

   The reference number is \(\pi/4\). Quadrant II has negative \(x\) and positive \(y\):

   \[ \boxed{(-\sqrt2/2,\sqrt2/2)}. \]

4. Find a coterminal value in \([0,2\pi)\) for \(25\pi/6\).

   Show Solution

   Subtract \(4\pi=24\pi/6\):

   \[ \boxed{\pi/6}. \]

5. Find the reference number for \(7\pi/5\).

   Show Solution

   The input is in Quadrant III, so subtract \(\pi\):

   \[ \frac{7\pi}{5}-\pi=\boxed{\frac{2\pi}{5}}. \]

6. Find the terminal point for \(-\pi/4\).

   Show Solution

   The coterminal input \(7\pi/4\) is in Quadrant IV:

   \[ \boxed{(\sqrt2/2,-\sqrt2/2)}. \]

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