Problem 65 - Algebra 2

Question

Sketch each angle in standard position. $$ 150^{\circ} $$

Step-by-Step Solution

Answer

The angle in standard position is a 150-degree rotation counterclockwise from the positive x-axis.

Step 1: Understand the concept of standard position

An angle is in standard position when its vertex is at the origin of a coordinate system and its initial side lines up with the positive x-axis. Rotation should be counterclockwise for positive angles and clockwise for negative angles.

Step 2: Draw a rough sketch

Start by drawing the x and y-axis and label them. This forms the Cartesian coordinate system. Draw a horizontal line (ray) starting from the origin pointing towards the right along the x-axis. This line is our initial side.

Step 3: Draw the angle

Rotate a line from the initial side counterclockwise by 150 degrees. This rotated line is the terminal side of the angle. The angle between the initial side and terminal side is our desired angle.

Step 4: Final sketch

\[The final sketch should look something like this:\n- An x and y axis intersecting at the origen (0,0)\n- A line (ray) starting from the origin and pointing to the right along the x axis (This is the initial side of the angle)\n- Another line (ray) starting from the origin and rotated 150 degrees counterclockwise from the initial side (This is the terminal side of the angle)\n- The space between these two rays is shaded to indicate the 150-degree angle\n\] Note that because our angle is greater than 90 but less than 180, the terminal side lies in the second quadrant.

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system, also known as the rectangular coordinate system, is a two-dimensional grid formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). This system helps in locating points in the plane using pairs of numbers, called coordinates, usually denoted as \((x, y)\).

• The intersection of these two axes is known as the origin, having coordinates \((0,0)\).
• It's crucial for plotting angles in standard position as it provides a clear framework to measure rotations around the origin.

Understanding this system is fundamental for sketching angles like 150° because it allows us to visualize rotations starting from the positive x-axis.

counterclockwise rotation

Counterclockwise rotation refers to the direction in which we typically measure positive angles from the positive x-axis, moving away from the initial side of an angle.

• In the context of the Cartesian coordinate system, this rotation is important when determining the terminal side of an angle.
• The standard convention is to rotate counterclockwise for positive angle measures, like the 150° in the exercise.

Counterclockwise movement is often used because of how it relates to the progression of angles and the assignment of quadrants in geometry.

second quadrant angle

The second quadrant is the section of the Cartesian coordinate system where:

• The x-values are negative, and the y-values are positive.
• Angles measure more than 90° but less than 180° when in standard position.

For an angle like 150°, its terminal side will land in the second quadrant. This placement significantly impacts calculations related to the angle, as trigonometric functions will have specific signs, e.g., sine will be positive while cosine and tangent will be negative.

terminal side of an angle

The terminal side of an angle in standard position is the ray which represents the angle's extent. It rotates from the initial side, positioned along the positive x-axis, by the specified angle measurement.

• In the example of 150°, you would start at the positive x-axis and rotate counterclockwise by 150°.
• This would place the terminal side in the second quadrant, as outlined above.

Identifying the terminal side is crucial because it helps to determine the exact placement of an angle within the coordinate system.