Problem 10 - Precalculus

Question

For the following exercises, draw an angle in standard position with the given measure. $$ -150^{\circ} $$

Step-by-Step Solution

Answer

Draw the angle in standard position by rotating clockwise 150 degrees from the positive x-axis.

Step 1: Understand Standard Position

An angle is said to be in standard position if its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. To draw an angle in standard position, we start from this initial side.

Step 2: Determine Direction of Angle

Since the angle measure is negative (-150°), the terminal side of the angle will be drawn rotating clockwise from the initial side on the positive x-axis.

Step 3: Visualize 150 Degrees Clockwise

Imagine the full circle divided into quadrants, each representing 90°. Since -150° exceeds -90° but is less than -180°, the angle will terminate in the third quadrant (bottom-left part of the circle).

Step 4: Draw the Angle

1. Start from the positive x-axis. 2. Rotate the terminal side clockwise passing through the second quadrant into the third quadrant. 3. Stop the terminal side at a position where you have completed a -150° rotation. The resulting angle should open towards the left-bottom quadrant.

Key Concepts

Standard Position

An angle is considered to be in "standard position" when its vertex is located at the origin of a coordinate system and its initial side points along the positive x-axis. This is a fundamental concept in trigonometry and helps create a uniform method to define angles.
Any angle can be measured by starting from this initial side and rotating around the origin to reach the terminal side.

• The vertex at the origin acts as the pivot for rotation.
• Standard position simplifies comparisons between different angles.
• It is an essential reference for understanding both positive and negative angles.

When solving problems involving angles, always ensure they are visualized or marked in standard position to align with mathematical conventions.

Negative Angles

In geometry, angles can be positive or negative. Their sign indicates the direction of the rotation from the initial side to the terminal side. Negative angles, like -150°, involve a clockwise rotation starting from the positive x-axis.
Understanding negative angles is key for solving many trigonometric problems. Here are some important points to remember:

• Negative angles rotate the terminal side clockwise.
• They are measured similarly to positive angles but in the opposite direction.
• Negative angle values continue to decrease as you move clockwise.

Imagine the angle -150° as starting from the positive x-axis and swinging towards the third quadrant in a clockwise direction.

Coordinate System

The coordinate system is the backdrop where angles in standard position are visualized. It's a grid formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The origin is the point of intersection, which serves as the starting point for angles.

• One of the most common coordinate systems is the Cartesian coordinate system.
• Coordinates like (x, y) help specify precise positions.
• The coordinate system is divided into four quadrants.

Understanding how angles are positioned within the coordinate system, using both the axes and the quadrant layout, is crucial for visualizing their characteristics accurately.

Quadrants

The plane in a coordinate system is divided into four distinct areas, or quadrants, based on the directionality of the x and y axes. Each quadrant offers different characteristics for positioning and measuring angles.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative (where -150° terminates).
• Quadrant IV: x is positive, y is negative.

The position of an angle, like -150°, can help determine which quadrant it lies in. Specifically, since -150° is beyond -90° but not past -180°, it falls into the third quadrant. This quadrant system helps in defining the characteristics of the terminal side of angles in a mathematically organized way.