Problem 42 - Algebra 2

Question

Find a positive and a negative coterminal angle for the given angle. $$ -425^{\circ} $$

Step-by-Step Solution

Answer

The positive coterminal angle is 295° and the negative coterminal angle is -785°.

Step 1: Finding the positive coterminal angle

Starting with the given angle of -425°, add 360° to get a positive coterminal angle. So we will calculate a = -425° + 360°

Step 2: Checking if the answer is positive

Check if the result from step 1 is positive. If it is not, keep adding 360° until a positive angle is obtained. Since -425° + 360° = -65°, which is not positive, we need to add 360° again. So, calculate a = -65° + 360°

Step 3: Finding the negative coterminal angle

Starting with the given angle of -425°, subtract 360° to get another negative coterminal angle. So b = -425° - 360°

Step 4: Checking if the result is negative

Check if the result from step 3 is negative. Since -425° - 360° = -785°, which is already negative, there is no need to subtract 360° again. Thus the negative coterminal angle is -785°.

Key Concepts

Positive Coterminal Angle

When we talk about coterminal angles, we are essentially dealing with angles that share the same initial and terminal sides. These angles can be recognized by adding or subtracting multiples of 360°.Finding a positive coterminal angle for a given angle, like -425°, involves making sure the result is above zero.

• Start with the given angle, -425°.
• Add 360° to get closer to a positive angle:\[ -425° + 360° = -65° \]
• -65° is still negative, so we need to add another 360°:\[ -65° + 360° = 295° \]

Thus, 295° is our positive coterminal angle for -425°.It's always good to verify that the final angle is indeed positive, and keeps up with the rotations that wrap the angle back into a positive measure.

Negative Coterminal Angle

Negative coterminal angles are those which, while possibly several full rotations away, still line up the same as the initial angle in question.If you start with an angle like -425°, one way to find a negative coterminal angle is by subtracting 360°.

• Begin with -425°.
• Subtract 360°, yielding:\[ -425° - 360° = -785° \]

The good thing about this result is that it's already negative.There's no need for further adjustment unless you aim for a different rotation, but -785° already shares the same sides as -425°.That's what makes them coterminal!

Angle Measurement

Understanding angle measurements is crucial because angles are fundamental in myriad areas, including geometry and trigonometry. Angles are commonly measured in degrees, which stems from dividing a circle into 360 degrees.

• Coterminal angles demonstrate how the measure of an angle can extend beyond the typical 0° to 360° range.
• The distinction among angles is established by the number and direction of 360° rotations applied to a starting angle.
• Positive rotations are counterclockwise, while negative rotations are clockwise.

In different scenarios, knowing how to convert and find coterminal angles helps in simplifying calculations and understanding angular movements. Using coterminal angles efficiently aids in problem-solving, especially in cyclic and periodic functions.