Problem 25 - Precalculus with Calculus Previews

Question

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=-5 \pi / 3 $$

Step-by-Step Solution

Answer

The reference angle is \(\frac{\pi}{3}\), \(\sin t = \frac{\sqrt{3}}{2}\) and \(\cos t = \frac{1}{2}\).

Step 1: Determine the Standard Position Angle

To find the reference angle, start by finding the angle in the standard position. Since the given angle is \(-\frac{5\pi}{3}\), which is negative, add \(2\pi\) (a full circle) to make it positive:\[t = -\frac{5\pi}{3} + 2\pi = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}.\]Thus, the angle in the standard position is \(\frac{\pi}{3}\).

Step 2: Find the Reference Angle

Since the angle \(\frac{\pi}{3}\) is already in the first quadrant, the reference angle \(t'\) is the angle itself. Therefore, the reference angle is:\[t' = \frac{\pi}{3}.\]

Step 3: Calculate Exact Values of \(\sin t \) and \(\cos t\)

Using the angle \(t = \frac{\pi}{3}\), find the exact values for \(\sin t\) and \(\cos t\). In the unit circle, these are known trigonometric values:\[\sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}, \quad \cos \left( \frac{\pi}{3} \right) = \frac{1}{2}.\]

Key Concepts

Understanding Reference Angles

Reference angles are a crucial part of trigonometry that help us determine the basic trigonometric values for any angle, regardless of its location on the unit circle. A reference angle is essentially the acute angle that a given angle makes with the x-axis.

To find the reference angle for an angle located in a standard position, follow these steps:

• First, adjust the given angle to fall within the range of 0 to 2π by either adding or subtracting full circles (2π), as necessary.
• Next, determine which quadrant the angle lies in. The reference angle is found by measuring the smallest angle needed to reach the x-axis. For angles in the first quadrant, the reference angle is the angle itself.

For example, with the angle t = -\frac{5\pi}{3}, we adjust it by adding 2\pi, resulting in \\(\frac{\pi}{3}\). This gives us a reference angle of \\(\frac{\pi}{3}\) since it already lies in the first quadrant.

Sine and Cosine Values on the Unit Circle

The unit circle is a powerful tool in trigonometry that helps us calculate sine and cosine values with ease. It is a circle centered at the origin (0,0)oh] with a radius of 1. The unit circle allows us to use the coordinates of points on the circle to find trigonometric values.

For any angle \(t\)in the unit circle:

• The x-coordinate of the point where the terminal side of the angle intersects the circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.

For example, consider the angle \(\frac{\pi}{3}\), which lies in the first quadrant of the unit circle. The coordinates at this angle are \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). Therefore, the sine \(\sin\left(\frac{\pi}{3}\right)\)is \(\frac{\sqrt{3}}{2}\), and the cosine \(\cos\left(\frac{\pi}{3}\right)\)is \(\frac{1}{2}\). These values are consistent for other angles that have a reference angle of \(\frac{\pi}{3}\), although their signs may differ depending on the quadrant.

Maximize Learning with the Unit Circle

Using the unit circle method is a fantastic way to enhance your understanding of trigonometric concepts like reference angles and sine and cosine values. On the unit circle, each angle relates to a point, making it easy to visualize and memorize key trigonometric values.

Key points to remember:

• The circle simplifies finding sine and cosine values, as angles correspond to coordinates \((x, y)\).
• Values repeat every \(2\pi\), reflecting the periodic nature of trigonometric functions.
• Examining angles and their terminal sides gives insight into trigonometric identities and angle relationships.

Building familiarity with the unit circle allows you to quickly deduce values for sine and cosine at various points. It is crucial to remember that each angle, along with its reference angle, can determine these values without the need for a calculator. This practice sharpens both your theoretical and practical understanding of trigonometry.