Problem 18 - Algebra 2 and Trigonometry

Question

In \(18-27,\) for each given angle, find a coterminal angle with a measure of \(\theta\) such that \(0 \leq \theta < 360\). $$ 390^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \( 30^{\circ} \).

Step 1: Identify the Problem

We need to find a coterminal angle for the given angle \( \theta = 390^{\circ} \). A coterminal angle differs from the original angle by integer multiples of \( 360^{\circ} \). We are looking for an angle that is between \( 0^{\circ} \) and \( 360^{\circ} \).

Step 2: Subtract 360 Degrees

Since \( 390^{\circ} \) is greater than \( 360^{\circ} \), we subtract \( 360^{\circ} \) to find a coterminal angle. This gives us: \[ 390^{\circ} - 360^{\circ} = 30^{\circ} \]

Step 3: Check the Resulting Angle

The result from Step 2 is \( 30^{\circ} \), which is within the range \( 0 \leq \theta < 360 \). Therefore, \( 30^{\circ} \) is the coterminal angle satisfying the condition.

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in geometry that helps us understand how far one line rotates around a fixed point to form another line. Angles are typically measured in degrees, with a full rotation being 360 degrees.
In the exercise provided, the angle given is 390 degrees. This indicates that we have a rotation that exceeds one complete circle since one complete circle is 360 degrees. Understanding angle measurement is essential because it allows us to describe and work with angles in a precise way.

• **Full Circle**: 360 degrees
• **Half Circle**: 180 degrees
• **Quarter Circle**: 90 degrees

Knowing these basic measurements helps us in manipulating and simplifying angles, especially in finding coterminal angles.

Subtracting Angles

When dealing with angles, especially those that exceed a complete circle, subtracting angles is a crucial step in finding coterminal angles. A coterminal angle is essentially any angle that ends at the same point on the circle as another angle, differing only by one or more full rotations.
In the example where the angle is 390 degrees, to return this large angle to a more manageable size within a standard circle, you subtract 360 degrees (one full circle): \[390^{\circ} - 360^{\circ} = 30^{\circ}\] This subtraction process makes sure that the angle lies within the standard 0 to 360-degree range. Subtracting or adding multiples of 360 degrees to bring an angle within this range is a common technique in trigonometry.

Angle Range

Understanding angle ranges is essential in determining which angles lie within a specified set of limits. In this case, we aim for the angle to lie between 0 and 360 degrees. This range represents a full circle, the standard range for expressing angles.
When asked to find a coterminal angle within this range, the goal is to adjust any given angle to fall between these limits using addition or subtraction of 360 degrees, which is the equivalent of a full circle rotation.

• 0 degrees: Represents no rotation from the starting point.
• 360 degrees: Represents a full rotation, equivalent to starting point.
• Angles like 390 degrees need adjusting to fit this range, resulting in 30 degrees after subtraction.

By ensuring the angle is within the 0 to 360-degree range, we're making it compatible with standard trigonometric functions and representations.