Question

Draw and label an angle in standard position with negative measure. Then, determine an angle with positive measure that is coterminal with your original angle. Show how to use a general expression for coterminal angles to find the second angle.

Short answer

The positive coterminal angle for -45 degrees is 315 degrees.

Step-by-step solution

Draw an Angle in Standard Position

First, draw an angle in standard position. This means the angle has its vertex at the origin of a coordinate plane and its initial side along the positive x-axis. For this exercise, let's choose an angle of -45 degrees.

Label the Angle

Label the drawn angle as -45 degrees. The negative sign indicates the angle is measured clockwise from the positive x-axis.

Determine Positive Coterminal Angle

To find an angle with a positive measure that is coterminal with the given angle, use the formula for coterminal angles: \[ \theta = \theta_0 + 360k \] where \theta_0\ is the given angle, and k is an integer. Set k=1 to get the smallest positive coterminal angle: \[ \theta = -45 + 360(1) = 315 \]

Verify the Coterminal Angle

Confirm that 315 degrees is coterminal with -45 degrees by checking that both angles share the same terminal side on the coordinate plane. They do, since 315 degrees is naturally found by rotating counterclockwise from the positive x-axis.

Key concepts

angle in standard position

An angle in standard position is defined by specific criteria. First, the vertex of the angle must be located at the origin of the coordinate plane. Second, the initial side of the angle should lie along the positive x-axis. This is a common starting point for measuring angles, whether they are positive or negative.
For instance, if we draw a -45 degree angle, we start from the positive x-axis and measure 45 degrees clockwise. This method is crucial for both drawing and understanding angles in various mathematical contexts.

negative angle measure

A negative angle measure indicates that the angle is measured clockwise from the positive x-axis. This is opposite to the direction used for positive angles, which are measured counterclockwise.

In our specific example, the angle is -45 degrees. We start at the positive x-axis and move 45 degrees in the clockwise direction. Negative angles can sometimes be confusing, but knowing that they simply denote a clockwise rotation helps clarify their meaning.

positive coterminal angle

A positive coterminal angle shares the same terminal side as a given angle but is measured in the counterclockwise direction. Being aware of coterminal angles is essential for solving many trigonometric problems.
To find a positive coterminal angle for a given negative angle, we can use the formula:
\[ \theta = \theta_0 + 360k \]
where \(\theta_0\) is the given angle, and \(k\) is an integer.
For a -45 degree angle, set \(k = 1\):
\[ \theta = -45 + 360 \times 1 = 315 \]
Thus, 315 degrees is the positive coterminal angle of -45 degrees.

general expression for coterminal angles

The general expression for coterminal angles helps find any angle that shares the same terminal side as the given angle. This is useful in various applications, such as simplifying angle measures or solving trigonometric equations.

The formula is:
\[ \theta = \theta_0 + 360k\]
where \(\theta_0\) is the initial angle and \(k\) is any integer. By changing the value of \(k\), we can find multiple coterminal angles. For example:

• For k = 0 (the original angle): \( \theta = -45 + 360 \times 0 = -45 \)
• For k = 1 (the smallest positive coterminal angle): \( \theta = -45 + 360 \times 1 = 315 \)
• For k = -1: \( \theta = -45 + 360 \times -1 = -405 \)

Understanding and using the general expression for coterminal angles makes it easier to navigate between positive and negative angle measures.