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The Special Right Triangles, \(30˚\)-\(60˚\)-\(90˚\) and \(45˚\)-\(45˚\)-\(90˚\), allow us to obtain exact values of the ordered pairs \((x, y)\) on a unit circle with standard angles \(30˚\), \(45˚\), or \(60˚\).

If we use symmetry across the \(y\)-axis and the \(x\)-axis, we can populate the known ordered pairs from QI into Quadrants II, III, and IV. Use the figures below to follow how this is done. To keep the pictures simple, \(30˚\) angles are marked, while \(45˚\) and \(60˚\) can be surmised by their larger magnitudes.

Notice the four \(30˚\) angles create a bow-tie look in Figure \(7.2.3\). These angles are called** reference angles.**

##### Definition: Reference Angle

Let \(\theta\) be a standard angle. A **reference angle**, denoted \(\hat{\theta}\), is the positive acute angle between the terminal side of \(\theta\) and the \(x\)-axis.

The word *reference* is used because all angles can *refer* to QI. That is, memorization of ordered pairs is confined to QI of the unit circle. If a standard angle \(\theta\) has a reference angle of \(30˚\), \(45˚\), or \(60˚\), the unit circle’s ordered pair is duplicated, but the sign value of \(x\) or \(y\) may need adjustment, depending on the quadrant of the terminal side of \(\theta\).

##### Example 7.2.1

The terminal side of a standard angle \(\theta = 225˚\) intersects the unit circle. State the ordered pair of the intersection.

**Solution**

The unit circle has radius \(r = 1\). Trigonometry weds algebra and geometry with visual sketches. We start with a sketch of the angle \(\theta = 225˚\). All standard angles begin on the positive-side of the \(x\)-axis. In which quadrant is the terminal side of \(\theta\)? That is, where does this angle come to a stop?

Where is the terminal side of \(\theta = 225˚\)? Since \(180˚ < 225˚ < 270˚\), the angle’s terminal side is in QIII.

The reference angle is computed by finding The difference between \(225˚\) and \(180˚\).

Note: The reference angle is never negative.

\(|225˚ − 180˚| = |180˚ − 225˚| = \hat{\theta}\)

\(45˚ = \hat{\theta}\)

In QIII, all ordered pairs \((x, y)\) are such that \(x < 0\) and \(y < 0\). Referring back to QI, using standard angle of \(45˚\) on the unit circle, the ordered pair \(\left( \dfrac{\sqrt{2}}{2} , \dfrac{\sqrt{2}}{2} \right)\) must be adjusted for negative \(x\) and \(y\) coordinates.

**Answer** The terminal side of \(\theta = 225˚\) intersects the unit circle at \(\left( −\dfrac{\sqrt{2}}{2} , −\dfrac{\sqrt{2}}{2} \right)\).

**Tip: **When a standard angle is greater than \(90˚\), use markers \(180˚\) or \(360˚\) to calculate the reference angle. Drawing a picture before computing is always recommended!

##### Example 7.2.2

The terminal side of a standard angle \(\theta = −480˚\) intersects the unit circle. State the ordered pair of the intersection.

**Solution**

Negative angles rotate clockwise. Sketch \(\theta\). Find a positive coterminal angle: \(−480˚ + 360˚(2) = 240˚\) Then apply the tips above or analyze visually: \(|180˚ − 240˚| = |−60˚| = 60˚ = \hat{\theta}\)

The ordered pair on the unit circle is \(\left(−\dfrac{1}{2} , −\dfrac{\sqrt{3}}{2} \right)\).

## Try It! (Exercises)

For #1-10, state the reference angle, \(\hat{\theta}\), of the given standard angle.

- \(\theta = 210˚ \)
- \(\theta = 350˚\)
- \(\theta = 110˚\)
- \(\theta = 240.5˚\)
- \(\theta = 142.75˚\)
- \(\theta = −315˚\)
- \(\theta = −230˚\)
- \(\theta = 500˚\)
- \(\theta = 615˚\)
- \(\theta = −835˚\)

For #11-20, the terminal side of the given standard angle, \(\theta\), intersects the unit circle at a point. State the ordered pair of the intersection.

- \(\theta = 135˚\)
- \(\theta = 300˚\)
- \(\theta = −240˚\)
- \(\theta = −150˚\)
- \(\theta = 420˚\)
- \(\theta = 570˚\)
- \(\theta = 840˚\)
- \(\theta = −765˚\)
- \(\theta = −930˚\)
- \(\theta = 1560˚\)

For #21-28, a standard angle’s rotation is described in words. You are given several hints about its rotation. Note: a full revolution is \(360˚\) (counter-clockwise) or \(-360˚\) (clockwise). **Find the measure of the angle described.**

- An angle has a counter-clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QIV. The reference angle \(\hat{\theta} = 30˚\).
- An angle has a counter-clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QII. The reference angle \(\hat{\theta} = 20˚\).
- An angle has a clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QIII. The reference angle \(\hat{\theta} = 10˚\).
- An angle has a clockwise rotation. The angle does not make a full revolution. The angle’s terminal side is in QII. The reference angle \(\hat{\theta} = 72˚\).
- An angle has a counter-clockwise rotation. The angle goes just beyond one full revolution. The angle’s terminal side is in QI. The reference angle \(\hat{\theta} = 55˚\).
- An angle has a counter-clockwise rotation. The angle goes beyond two full revolutions. The angle’s terminal side is in QII. The reference angle \(\hat{\theta} = 24˚\).
- An angle has a clockwise rotation. The angle goes beyond one full revolution. The angle’s terminal side is in QIII. The reference angle \(\hat{\theta} = 18˚\).
- An angle has a clockwise rotation. The angle goes beyond two full revolutions. The angle’s terminal side is in QIV. The reference angle \(\hat{\theta} = 39˚\).