Problem 39 - Algebra and Trigonometry

Question

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°.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and final sides but have different rotations. To find an angle coterminal with a given angle, we can add or subtract multiples of 360°, as 360° represents a full circle.

Step 2: Finding Coterminal Angle

Start with the given angle, 733°. To find an angle coterminal with 733°, subtract 360° repeatedly until the result is between 0° and 360°.

Step 3: First Subtraction

Subtract 360° from 733°: \[ 733° - 360° = 373° \] 373° is still greater than 360°, so we need to subtract 360° once more.

Step 4: Second Subtraction

Subtract 360° from 373°: \[ 373° - 360° = 13° \] Since 13° is between 0° and 360°, this is our coterminal angle.

Step 5: Conclusion

The angle 13° is coterminal with 733° and lies within the range of 0° to 360°.

Key Concepts

Angle Subtraction

Finding a coterminal angle involves subtracting or adding a full circle until the angle is between 0° and 360°. For the given angle of 733°, we need to subtract 360° multiple times. This repeats because each 360° represents one complete revolution around a circle.

• Start with 733° and subtract 360°, yielding 373°.
• Once again, subtract 360° from 373°, resulting in 13°.

By doing these subtractions, we effectively "unwrap" the given angle back to a standard position within one rotation of the circle.

Full Circle

A full circle in trigonometry is equivalent to 360 degrees. This concept is crucial for understanding coterminal angles. When dealing with angles, one full revolution corresponds to adding or subtracting 360 degrees. This process helps in reducing any angle to a simpler form.

• Think of the circle like a clock, where going all the way around brings you back to where you started.
• If an angle exceeds 360°, imagine continuing around the circle for extra rotations.

Using this idea, we can always find an equivalent angle within the first complete rotation (0° to 360°).

Trigonometry

Trigonometry is the study of angles and their relationships. It's essential for understanding how angles work in circles. In the context of coterminal angles, trigonometry helps us simplify angles by considering them in terms of their standard positions.

• Angles, when represented in circles, offer new ways of understanding and solving problems.
• Trigonometric functions rely on the concept of angles, which is why simplifying them is critical.
• Coterminal angles maintain the same trigonometric values because they share identical terminal sides on the circle.

Angle Between 0 and 360 Degrees

The goal in finding coterminal angles is to adjust the given angle so it falls between 0° and 360°. This range is called the "standard position," which helps in easily evaluating trigonometric characteristics of angles.

• An angle in this range is more manageable and straightforward for calculations.
• For the problem with 733°, after subtracting 360° twice, we achieved 13°.
• This angle, 13°, is ready for use in any trigonometric calculations or representations.

Ultimately, positioning the angle within this boundary simplifies many mathematical analyses and applications.