Problem 25 - Algebra and Trigonometry

Question

\(25-30\) me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ 50^{\circ} $$

Step-by-Step Solution

Answer

Positive: \(410^{\circ}, 770^{\circ}\); Negative: \(-310^{\circ}, -670^{\circ}\).

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. To find coterminal angles, we add or subtract multiples of a full circle, which is \(360^{\circ}\).

Step 2: Find the First Positive Coterminal Angle

To find a positive coterminal angle, add \(360^{\circ}\) to the given angle: \( 50^{\circ} + 360^{\circ} = 410^{\circ} \). Hence, \(410^{\circ}\) is a positive coterminal angle with \(50^{\circ}\).

Step 3: Find the Second Positive Coterminal Angle

Add another \(360^{\circ}\) to \(410^{\circ}\), giving us \( 410^{\circ} + 360^{\circ} = 770^{\circ} \). Thus, \(770^{\circ}\) is the second positive coterminal angle.

Step 4: Find the First Negative Coterminal Angle

To find a negative coterminal angle, subtract \(360^{\circ}\) from the given angle: \( 50^{\circ} - 360^{\circ} = -310^{\circ} \). Hence, \(-310^{\circ}\) is a negative coterminal angle.

Step 5: Find the Second Negative Coterminal Angle

Subtract another \(360^{\circ}\) from \(-310^{\circ}\), resulting in \( -310^{\circ} - 360^{\circ} = -670^{\circ} \). This gives us the second negative coterminal angle, \(-670^{\circ}\).

Key Concepts

Standard Position

When we talk about angles in their "standard position", we are referring to a specific way of drawing angles on the coordinate plane. An angle is said to be in standard position when:

• The vertex is at the origin of the coordinate plane, which is the point (0, 0).
• The initial side of the angle lies along the positive x-axis.

As we measure the angle, it is formed by rotating a ray (the terminal side) around the origin. It's important to note that angles can be measured in degrees or radians, but in this context, we're focusing on degrees. Seeing angles in standard position helps us understand and visualize their direction and magnitude easily. It also forms the basis for finding coterminal angles.

Positive Angles

Positive angles are measured in a counterclockwise direction from the initial side of the angle to the terminal side. This is the most common way to measure angles, as the counterclockwise movement is typically considered the "positive" direction. Whenever we need to find a positive coterminal angle:

• Add a full circle (360°) to the given angle.
• Repeat the addition to find multiple positive coterminal angles.

For example, given a 50° angle, a positive coterminal angle would be 50° + 360° = 410°, and another would be 410° + 360° = 770°. These angles share the same terminal side as the original 50° but have been rotated one or more full circles.

Negative Angles

Negative angles are measured in a clockwise direction, which is considered the opposite of the standard positive direction. They make for great practice in understanding angle positioning and rotation on the circle. To find negative coterminal angles, you can:

• Subtract a full circle (360°) from the given angle.
• Continue subtracting to find more negative coterminal angles if needed.

For instance, if you start with 50°, to find a negative coterminal angle, calculate 50° - 360° = -310°. To find another, simply repeat: -310° - 360° = -670°. These angles also share the same terminal side as the original angle.

Full Circle

A "full circle" in angle measurement terms refers to a complete rotation of 360°. It represents one complete wrap around a circle's circumference. This concept is essential when determining coterminal angles. A full circle is helpful because:

• Adding or subtracting 360° to an angle results in the angle ending at the same terminal position.
• It allows us to find the coterminal angles easily, by simple calculations based on 360° rotations.

Remember, regardless of the number of full circles added or subtracted, the angles will still point to the same position, hence qualifying them as coterminal. This makes the full circle concept a vital tool in solving angle-related problems.