Problem 35 - Precalculus Mathematics for Calculus

Question

Find the reference number for each value of \(t.\) (a) \(t=\frac{5 \pi}{4}\) (b) \(t=\frac{7 \pi}{3}\) (c) \(t=-\frac{4 \pi}{3}\) (d) \(t=\frac{\pi}{6}\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{4}\), (b) \(\frac{\pi}{3}\), (c) \(\frac{\pi}{3}\), (d) \(\frac{\pi}{6}\)

Step 1: Understanding Reference Numbers

Reference numbers, or reference angles, are always positive and within the first quadrant, specifically between 0 and \(\frac{\pi}{2}\) radians. To find the reference number for \(t\), you need to understand in which quadrant \(t\) lies or convert it to a positive angle first.

Step 2: Find Reference Number for (a) \(t=\frac{5\pi}{4}\)

Since \(\frac{5\pi}{4}\) radians lies in the third quadrant which is \(\pi < t < \frac{3\pi}{2}\), the reference angle is \(t - \pi = \frac{5\pi}{4} - \pi = \frac{\pi}{4}\).

Step 3: Find Reference Number for (b) \(t=\frac{7\pi}{3}\)

\(\frac{7\pi}{3}\) is an angle larger than \(2\pi\). We first find an equivalent angle within \(0\) to \(2\pi\) by calculating: \(\frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}\). This means \(\frac{7\pi}{3}\) has a reference number \(\frac{\pi}{3}\).

Step 4: Find Reference Number for (c) \(t=-\frac{4\pi}{3}\)

To find a positive angle, add \(2\pi\) to \(-\frac{4\pi}{3}\): \(-\frac{4\pi}{3} + \frac{6\pi}{3} = \frac{2\pi}{3}\). Since \(\frac{2\pi}{3}\) is in the second quadrant, its reference angle is \(\pi - \frac{2\pi}{3} = \frac{\pi}{3}\).

Step 5: Find Reference Number for (d) \(t=\frac{\pi}{6}\)

\(\frac{\pi}{6}\) is already an acute angle within the first quadrant, so its reference number is \(\frac{\pi}{6}\) itself.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between angles and sides of triangles. This area of math plays a crucial role in analyzing and solving problems involving periodic phenomena. In trigonometry, you'll encounter different functions, such as sine, cosine, and tangent, which relate to angles in triangles. These functions can describe phenomena like sound and light waves, circuits, and oscillating systems.

When dealing with angles, reference angles come into play. These are always positive and help simplify problems by converting angles to their basic form within the first quadrant. They give us a standard angle measure that makes calculations more straightforward. Understanding how reference angles work is fundamental in trigonometry for solving real-world problems and analyzing scientific data.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians give a more natural approach by relating the arc length of a circle directly to the radius. One full rotation around a circle, equivalent to 360 degrees, is expressed as \(2\pi\) radians.

To convert between degrees and radians, you can use the fact that \(360^\circ = 2\pi\) radians or \(180^\circ = \pi\) radians. For example, an angle of \(90^\circ\) is equivalent to \(\frac{\pi}{2}\) radians. Understanding radians is vital because this measurement system is commonly used in more advanced mathematics and physics.

When working with reference angles or calculating trigonometric values, you'll often find yourself converting angles to radians for easier and more precise calculations.

Angle Conversion

Angle conversion involves changing an angle from one unit of measure to another, typically from degrees to radians or vice versa. This is important in trigonometry since different applications might require angles in specific units.

To convert degrees to radians, multiply the number of degrees by \(\frac{\pi}{180}\). Conversely, to convert radians to degrees, multiply the radians by \(\frac{180}{\pi}\). For example, to convert \(45^\circ\) to radians, calculate \(45 \times \frac{\pi}{180} = \frac{\pi}{4}\) radians.

Considering quadrant locations is another aspect of understanding angles. Each angle can be identified in one of the four quadrants of the coordinate plane, and knowing this helps in finding reference angles, which are fundamental in solving trigonometric equations.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Using the unit circle, you can define trigonometric functions like sine and cosine for all real numbers. Each point on the unit circle corresponds to an angle in radians, providing the x-coordinate as the cosine of the angle and the y-coordinate as the sine.

This makes the unit circle an extremely efficient tool for understanding trigonometric identities and the properties of these functions as they relate to angles. As you circle around the origin, the reference angles are reflections over the x or y axes to simplify calculations.

Mastering the unit circle concept leads to a deeper understanding of how trigonometric functions behave and helps solve various math problems, especially those involving cyclic patterns or harmonic motion.