Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the reference angle \(\theta^{\prime}\) for the special angle \(\theta .\) Sketch \(\theta\) in standard position and label \(\boldsymbol{\theta}^{\prime}\). $$\theta=315^{\circ}$$

Step-by-Step Solution

Answer

The reference angle \(\theta^{\prime}\) for the special angle \(\theta=315^{\circ}\) is \(\theta^{\prime}=45^\circ\).

Step 1: Understanding what is a reference angle

A reference angle \(\theta^{\prime}\) is the acute angle that the terminal side of the given angle \(\theta\) makes with the x-axis. It is always positive and its magnitude will be between 0° and 90°.

Step 2: Finding the reference angle

Given special angle \(\theta=315^{\circ}\). This angle terminates in the fourth quadrant. For angles that lie in the fourth quadrant, the reference angle is calculated as \(\theta^{\prime} = 360^\circ - \theta\). Thus, \(\theta^{\prime} = 360^\circ - 315^\circ = 45^\circ\)

Step 3: Sketching the angle and reference angle

To sketch the angle, a horizontal line is drawn to represent x-axis and a vertical line for y-axis. From the origin, the line rotates in an anti-clockwise direction until reaching the terminal side of \(\theta=315^\circ\). The reference angle \(\theta^{\prime}\) is then drawn which is the angle from the terminal side of \(\theta\) and x-axis.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics focused on the relationships between angles and sides of triangles, especially right-angled triangles. By extending these concepts, trigonometry is utilized to explore the properties and behaviors of any angle, their rotations, and how they interact with the coordinate system. In particular, the concept of a reference angle demonstrates the usefulness of trigonometry in simplifying complex angle measures into more manageable acute angles that fall within a 0° to 90° range.

Special Angles

Special angles in trigonometry refer to angles that have readily recognizable sine, cosine, and tangent values, often memorized for their prevalence in mathematics. These are typically angles like 30°, 45°, and 60°. In the context of a reference angle, the advantage of these special angles becomes apparent as they are easily compared to their reference angles in the coordinate plane, simplifying calculations and understanding of trigonometric functions.

Terminal Side

The terminal side is the ray that extends from the angle's vertex, usually positioned at the origin of a coordinate plane, and passes through a point that determines the specific measure of the angle. When considering angles greater than 360° or less than 0°, the terminal side is defined after the angle has completed the necessary rotations. It is crucial in defining both the initial angle and its reference angle, as the reference angle is formed by the acute angle between the terminal side and the closest x-axis direction.

Standard Position

An angle is said to be in standard position when its vertex is at the origin of a coordinate plane and one ray, known as the initial side, is positioned on the positive x-axis. The other ray, the terminal side, is rotated from the initial side to the designated angle measure. Standard position is significant for establishing a common frame of reference, which allows for the consistent determination of reference angles and application of trigonometric functions across various quadrants.