Problem 48 - Precalculus: A Concise Course

Question

Find the reference angle \(\boldsymbol{\theta}^{\prime}\), and sketch \(\boldsymbol{\theta}\) and \(\boldsymbol{\theta}^{\prime}\) in standard position. $$ \theta=-215^{\circ} $$

Step-by-Step Solution

Answer

The reference angle \(\theta'\) is 35 degrees.

Step 1: Convert Given Angle to Positive

Given angle \(\theta\) is -215 degrees. Since negative angle direction is clockwise and positive angle direction is counter-clockwise, start from the positive x-axis and move 215 degrees clockwise. This gives \(\theta\) in standard position. To work with a positive angle, add 360 degrees to \(\theta\) to get the equivalent positive angle. \( \theta + 360^{\circ} = -215^{\circ} + 360^{\circ} = 145^{\circ} \).

Step 2: Determine Reference Angle

The reference angle, \(\theta'\), is the acute angle that the terminal side of \(\theta\) makes with the x-axis. Since the positive angle is in the second quadrant the reference angle will be \(180^{\circ} - \theta = 180^{\circ} - 145^{\circ} = 35^{\circ}\). So, \(\theta' = 35^{\circ}\).

Step 3: Sketch θ and θ' in Standard Position

Draw a coordinate plane either on paper or using a graphing tool. Start at the positive x-axis to keep the initial side. Rotate 215 degrees in the clockwise direction to delineate \(\theta\). From the terminal side of \(\theta\), draw a line to the x-axis to indicate \(\theta'\).

Key Concepts

standard position

When we talk about angles in a mathematical or geometric context, using the term "standard position" is quite common. An angle is considered to be in the standard position when its vertex is placed at the origin of a Cartesian coordinate plane, and its initial side lies along the positive x-axis.

This position allows us to measure angles in both the counter-clockwise and clockwise directions. Typically, counter-clockwise measurements are considered positive, while clockwise measurements are considered negative.

• **Positive Angles:** Measured counter-clockwise from the positive x-axis.
• **Negative Angles:** Measured clockwise from the positive x-axis.

By positioning an angle this way, we can easily visualize and grasp its size and relationship with other angles or figures. Thus, understanding its orientation is key to identifying the angle's impact on graphs and equations.

acute angle

An acute angle is one which measures less than 90 degrees. It is known for being sharp and small in comparison to other types of angles, like right or obtuse angles. In the context of finding reference angles, the acute angle serves as the smallest angle formed between the terminal side of a given angle and the x-axis.

Reference angles are always positive and are used to facilitate trigonometric calculations by simplifying the problem, basically capturing the essence of the angle's position. This angle is also beneficial when comparing it across different quadrants:

• Allows for the use of symmetrical properties.
• Helps simplify calculations by reducing problem complexity.

An acute angle, due to its small size, plays a pivotal role in simplifying trigonometric assessments.

quadrantal angles

Quadrantal angles are angles whose terminal sides lie directly along the x-axis or y-axis. These angles are critical benchmarks in trigonometry and are identified by their multiples of 90 degrees (such as 0°, 90°, 180°, 270°, 360°, etc.).

Knowing when an angle is quadrantal makes it easier to determine trigonometric functions like sine, cosine, and tangent because they have specific, fixed values at these angles. These fixed values make calculations straightforward:

• **0° and 180°:** Angles where the terminal side aligns with the x-axis.
• **90° and 270°:** Angles where the terminal side aligns with the y-axis.

These particular positions do not form acute angles with the x-axis, as their sides coincide directly with the axes, providing instantaneous and clear data without further breakdown.