Problem 41 - Algebra and Trigonometry

Question

Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 1110^{\circ} $$

Step-by-Step Solution

Answer

The angle is 30°.

Step 1: Convert Negative to Positive

Coterminal angles can be found by adding or subtracting full rotations around the circle, which are multiples of 360 degrees. We need an angle between 0 and 360 degrees, so let's find an equivalent positive angle by subtracting 360 until we are in this range.

Step 2: Subtract 360° Once

Subtract 360° from 1110°.\[1110° - 360° = 750°\]750° is still greater than 360°, so we need to subtract 360° again.

Step 3: Subtract 360° Again

Subtract 360° from 750° again.\[750° - 360° = 390°\]390° is still greater than 360°, so we need to subtract 360° once more.

Step 4: Subtract 360° Once More

Subtract 360° from 390° one last time to find the angle under 360°.\[390° - 360° = 30°\]30° is now within our target range of 0° to 360°.

Key Concepts

Angle Conversion

Understanding how to convert angles between different units or ranges is crucial in trigonometry. When we talk about "angle conversion," we often deal with converting between degrees and radians or finding coterminal angles within a specific range.
For example, degrees are typically used in geometry and daily life, while radians are used in higher mathematics and physics.

• To convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\).
• To convert from radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

In our problem, we are not converting between degrees and radians but finding coterminal angles. This means adjusting angles by a full circle (360°) to find an equivalent angle in a specified range, such as between 0° and 360°.

Full Rotations

Full rotations refer to circling completely around a point or axis, covering 360 degrees or \(2\pi\) radians. Understanding full rotations is important when finding coterminal angles.
Coterminal angles are angles that share the same terminal side after one or more full rotations.

• A positive full rotation is +360°.
• A negative full rotation is -360°.

To find coterminal angles, you can add or subtract 360° as many times as needed until the angle lies within the desired range.
In our example, starting from 1110°, multiple full rotations were subtracted:

• First, subtracting once yields 750°.
• Subtracting again results in 390°.
• A third subtraction finally reduces it to 30°.

The angle 30° is a coterminal angle that occurs after several full rotations.

Trigonometry Basics

Trigonometry is a branch of mathematics dealing with angles, triangles, and trigonometric functions. Foundational concepts like coterminal angles are essential in understanding trigonometry.
Key trigonometric functions include sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides. Each function has a specific role:

• The sine function compares the opposite side to the hypotenuse.
• The cosine function compares the adjacent side to the hypotenuse.
• The tangent function compares the opposite side to the adjacent side.

By understanding coterminal angles and their reductions, students can better grasp how these functions behave periodically and how they repeat every full 360° rotation or 2\(\pi\) radians. This knowledge aids in graphing and solving trigonometric equations that involve periodic behavior.