Question

Draw the angle \(\frac{5 \pi}{6}\) in standard position.

Short answer

Draw \(\frac{5 \pi}{6}\) in the second quadrant starting from the positive x-axis.

Step-by-step solution

- Understanding the Problem

The angle \(\frac{5 \pi}{6}\) needs to be drawn in standard position. Standard position means the angle's initial side is on the positive x-axis.

- Convert to Degrees (Optional for Understanding)

To get a better sense of what \(\frac{5 \pi}{6}\) radians translates to in degrees, use the conversion factor \(\frac{180^\text{o}}{\pi}\). Compute \(\frac{5\pi}{6} \times \frac{180^\text{o}}{\pi} = 150^\text{o}\).

- Identify the Quadrant

Since \(\frac{5 \pi}{6}\) is equivalent to 150 degrees, note that it lies in the second quadrant because 150 degrees is between 90 degrees and 180 degrees.

- Draw the Angle

Start by drawing a coordinate plane. Place the initial side of the angle on the positive x-axis. Measure an angle of 150 degrees counterclockwise starting from the positive x-axis. The terminal side of the angle will be in the second quadrant.

- Finalize the Drawing

Label the angle \(\frac{5 \pi}{6}\) on the coordinate plane, ensuring the initial side is on the positive x-axis and the terminal side is correctly positioned in the second quadrant.

Key concepts

Standard Position

In trigonometry, placing an angle in standard position makes it easier to understand and visualize.
The standard position of an angle has its initial side on the positive x-axis.
From here, you measure counterclockwise for positive angles and clockwise for negative angles.
This standard reference helps in clearly identifying where the angle's terminal side lands on the coordinate plane.
Imagining or drawing the standard position of angles is a foundational skill, aiding in various applications like plotting on graphs or solving trigonometric equations.

Radian to Degree Conversion

Radians and degrees are two units for measuring angles.
Converting between these is crucial for understanding and solving many trigonometric problems.
The conversion factor between radians and degrees is \(\frac{180^\text{o}}{\text{π}}\).
To convert an angle from radians to degrees, multiply the radian measure by this factor.
For instance, converting \(\frac{5\text{π}}{6}\) to degrees involves: \(\frac{5\text{π}}{6} \times \frac{180^\text{o}}{\text{π}} = 150^\text{o}\).
Understanding both units and being fluent in converting between them enhances problem-solving skills.

Quadrant Identification

Identifying the quadrant where an angle's terminal side lies is crucial.
The coordinate plane is divided into four quadrants:

• First Quadrant: 0 to 90 degrees
• Second Quadrant: 90 to 180 degrees
• Third Quadrant: 180 to 270 degrees
• Fourth Quadrant: 270 to 360 degrees.

For the angle \(\frac{5\text{π}}{6}\) or 150 degrees, it falls between 90 and 180 degrees, placing it in the second quadrant.
Knowing the quadrant helps in determining the signs of sine, cosine, and tangent values.

Coordinate Plane

The coordinate plane is a foundational concept for graphing angles, functions, and shapes.
It's a two-dimensional plane where each point is defined by a pair of numerical coordinates \((x, y)\).
Divided into four quadrants by the x-axis (horizontal) and y-axis (vertical):

• First Quadrant: \((+,+)\)
• Second Quadrant: \((- \frac{1}{2}, 0)\)
• Third Quadrant: \((-,-)\)
• Fourth Quadrant: \((+,-)\)

Understanding this layout is essential for plotting angles. In our example, to draw \(\frac{5\text{π}}{6}\), start from the positive x-axis and measure 150 degrees counterclockwise.
The terminal side will land in the second quadrant, accurately reflecting the angle's position.