Problem 13 - Precalculus : Building Concepts and Connections

Question

Skills This set of exercises will reinforce the skills illustrated in this section. In Exercises \(9-22,\) find the reference angle for each of the angles given. $$\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The reference angle for \(\frac{3\pi}{4}\) is \(\frac{\pi}{4}\).

Step 1: Identify Quadrant

First, remember that the angle \(\frac{3\pi}{4}\) lies within the second quadrant because \(\frac{\pi}{2} < \frac{3\pi}{4} < \pi\).

Step 2: Apply Formula

Next, apply the formula for the reference angle in the second quadrant which is \(\pi - \text{angle}\). Substituting \(\frac{3\pi}{4}\) into the formula gives \(\pi - \frac{3\pi}{4}\).

Step 3: Calculate

Doing the calculation getting \(\frac{\pi}{4}\) as a result.

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics that deals with the relationships between lengths and angles of triangles, particularly the right triangles. An essential element of trigonometry is the concept of an angle, which is typically measured in degrees or radians. When working with trigonometry in the context of the unit circle, radians are often the preferred unit of measurement because they provide a direct connection between the angle measure and arc length on the unit circle.

One quirky aspect of trigonometry is the use of reference angles. A reference angle is the acute angle formed by the terminal side of an angle in standard position and the closest axis (x-axis or y-axis). Every angle in standard position (where one ray of the angle is on the positive x-axis) has a reference angle, which is always between 0 and \( \pi/2 \) radians (or 0 and 90 degrees). Reference angles are used to simplify the process of finding trigonometric values for angles that lie in any quadrant of the unit circle.

Quadrants of the Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. It’s a crucial tool in trigonometry because it helps to understand the sine, cosine, and tangent functions. The circle is divided into four quadrants by the x-axis and y-axis. Each quadrant contains a range of angles, each with unique properties for the trigonometric functions.

Quadrant I

Angles from 0 to \( \pi/2 \) radians. Both sine and cosine are positive.

Quadrant II

Angles from \( \pi/2 \) to \( \pi \) radians. Sine is positive, while cosine is negative.

Quadrant III

Angles from \( \pi \) to \(3\pi/2\) radians. Both sine and cosine are negative.

Quadrant IV

Angles from \(3\pi/2\) to \(2\pi\) radians. Cosine is positive, while sine is negative.

Knowing which quadrant an angle resides in is crucial when calculating reference angles because it determines how the reference angle is found relative to the axis.

Angle Calculation

Calculating an angle, specifically a reference angle, involves understanding both the measure of the angle and its position in the unit circle. The calculation method varies depending on which quadrant the angle lies in, since the reference angle is always the smallest angle that can be made from the terminal side of the original angle to the x-axis.

In the first quadrant, the reference angle is the angle itself as it is already acute. In the second and third quadrants, the reference angle is the difference between \( \pi \) radians (or 180 degrees) and the angle. In the fourth quadrant, it is the difference between \(2\pi\) radians (or 360 degrees) and the angle.

As for the original exercise \( \frac{3 \pi}{4} \) that lies in the second quadrant, to find its reference angle we subtract it from \( \pi \) radians. This yields the result \( \pi - \frac{3 \pi}{4} = \frac{\pi}{4} \) radians. This calculation is key to finding the trigonometric functions' values for angles not located in the first quadrant but using their reference angles which have the same sine, cosine, and tangent values, but possibly different signs depending on the quadrant.