Problem 37 - Algebra and Trigonometry

Question

\(37-42\) me Find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with the given angle. $$ 733^{\circ} $$

Step-by-Step Solution

Answer

The coterminal angle is \(13^{\circ}\).

Step 1: Identify the Range

We need to find an angle between \(0^{\circ}\) and \(360^{\circ}\) that is coterminal with \(733^{\circ}\). Coterminal angles differ by multiples of \(360^{\circ}\).

Step 2: Subtract 360°

Start by subtracting \(360^{\circ}\) from \(733^{\circ}\). Calculate: \(733^{\circ} - 360^{\circ} = 373^{\circ}\).

Step 3: Subtract 360° Again

Since \(373^{\circ}\) is still greater than \(360^{\circ}\), subtract \(360^{\circ}\) again. Calculate: \(373^{\circ} - 360^{\circ} = 13^{\circ}\).

Step 4: Verify the Range

Now that we have \(13^{\circ}\), check if this angle is within \(0^{\circ}\) to \(360^{\circ}\). It is between the given range, so this is the coterminal angle.

Key Concepts

Coterminal Angles

Coterminal angles are quite fascinating in the world of trigonometry and angles. An angle is said to be coterminal with another if both can be drawn on the same position on the circle. This means they share both the origin and the direction of their terminal sides.
A neat trick with coterminal angles is that they differ by full rotations, which in degrees is represented by a multiple of 360°.

• For example, the angles 30° and 390° are coterminal because 390° - 360° = 30°.
• In general, you can find multiple coterminal angles for a given angle by adding or subtracting 360° repeatedly.

In practice, this concept allows us to convert angles outside the standard range, typically between 0° and 360°, into an equivalent angle that is easy to work with.
Being able to quickly recognize or calculate coterminal angles can be incredibly helpful for solving problems associated with rotations and symmetry. 🔄

Angle Measurement

When talking about angles, it's all about how they open up from a common point—usually labeled the vertex. Angles are usually measured in degrees (°).
These measurements help us understand how wide that opening is. A full circle comprises 360°, offering a systematic way to analyze different rotations or portions of a circle.

• A quarter circle is 90°, which we relate to each quadrant in the unit circle.
• A half circle is 180°, often called a straight angle.
• Angles greater than 360° depict more than one full rotation around the circle.

Knowing how to move flexibly within the given range of 0° to 360° is key to many calculations and graph interpretations in trigonometry. Making sense of angles outside this range by converting them helps us maintain this standardized framework for easier computation and understanding.

Subtracting Multiples of 360 Degrees

Finding an angle within a specific range, like the typical 0° to 360°, often requires a bit of adjustment if the original angle exceeds this range. One key modification is subtracting multiples of 360° to create an equivalent and manageable angle.
Here's how this works. If you start with an angle, such as 733°, check if it's beyond 360°. If so, subtract 360° once and see where you end up.

• If it's still above 360°, simply subtract 360° again until you get a number within the desired range.
• For instance, for 733°, subtracting 360° gives us 373°, and then subtracting 360° again gives us 13°, a handy angle.

This technique of repeatedly subtracting 360° isn't just a methodical procedure; it's practical! You maintain the angle's coterminal status while also fitting it neatly within a more familiar and workable range. It facilitates easier calculation, drawing, and application in broader trigonometric problems.