Problem 42 - Algebra and Trigonometry

Question

Find (a) the reference number for each value of t, and (b) the terminal point determined by t. $$ t=-\frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

Reference angle: \(\frac{\pi}{6}\), Terminal point: \((-\frac{\sqrt{3}}{2}, -\frac{1}{2})\).

Step 1: Find the Reference Angle

To find the reference angle for a given angle in radians, first identify which quadrant the angle lies in. The angle \(-\frac{7\pi}{6}\) is in the third quadrant as it is between \(-\pi\) and \(-\frac{3\pi}{2}\). The reference angle for an angle \(t\) in the third quadrant can be found using: \(\text{Reference Angle} = |t - \pi|\). Calculate: \(|-\frac{7\pi}{6} - (\pi)| = |-\frac{7\pi}{6} + \frac{6\pi}{6}| = \frac{\pi}{6}\).

Step 2: Calculate the Terminal Point

The terminal point for an angle parameter \(t\) on the unit circle can be found using the coordinates \((\cos(t), \sin(t))\). Given \(t = -\frac{7\pi}{6}\), we use the reference angle \(\frac{\pi}{6}\), for which \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) and \(\sin(\frac{\pi}{6}) = \frac{1}{2}\). Since \(t\) is in the third quadrant, both cosine and sine should be negative. Thus, the terminal point is \((-\frac{\sqrt{3}}{2}, -\frac{1}{2})\).

Key Concepts

Reference Angle

A reference angle is the smallest angle that a given angle makes with the x-axis. It is always a positive angle, measured in the range from 0 to \ \( \frac{\pi}{2} \) \ (or 0° to 90° in degrees).Here's how to find it for angles in different quadrants:

• **First Quadrant:** The reference angle is the same as the actual angle, since it already measures from the x-axis.
• **Second Quadrant:** Subtract the angle from \( \pi \) (or 180°).
• **Third Quadrant:** Subtract \( \pi \) from the angle (or subtract 180°).
• **Fourth Quadrant:** Subtract the angle from \( 2\pi \) (or 360°).

For our exercise with \(-\frac{7\pi}{6}\), it lies in the third quadrant. Therefore, the reference angle is: \[ | -\frac{7\pi}{6} - \pi | = \frac{\pi}{6} \]The reference angle provides a simple basis for calculating trigonometric functions like sine and cosine.

Terminal Point

The terminal point on the unit circle corresponds to the coordinates where an angle \( t \) lands on the circle. This is crucial because from this point, we can directly determine the values of trigonometric functions. To locate this point for any angle, we use the formulas:

• Cosine corresponds to the x-coordinate.
• Sine corresponds to the y-coordinate.

For \( t = -\frac{7\pi}{6} \), we first find the reference angle, which is \( \frac{\pi}{6} \). We know:

• \( \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \)
• \( \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \)

Since our angle is in the third quadrant, both cosine and sine values are negative. Therefore, the terminal point is:\[\left( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right)\]This negative sign reflects the direction on the unit circle.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin (0,0) of the coordinate system. This circle allows us to define all sine and cosine values. Key aspects of the unit circle include:

• **Radius**: Always 1.
• **Center**: The origin, or point (0,0).
• **Circumference**: Covered as you move around, represented by angles in radians or degrees.

Angles are measured starting from the positive x-axis.A full circle is \( 2\pi \) radians (or 360°). As you can imagine, finding the terminal point means determining where an angle ends up on this circle.Using the unit circle simplifies complex trigonometric calculations. The x-coordinate of a terminal point gives the cosine of an angle, while the y-coordinate gives the sine. The unit circle's consistency ensures that these outputs remain between -1 and 1 for all angles, which beautifully fits all trigonometric functions like sine and cosine into an understandable model. Using the unit circle allows us to visualize and compute the trigonometric values needed to solve problems effortlessly.