Problem 21 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Sketch the angle. Then find its reference angle. \(-\frac{5 \pi}{6}\)

Step-by-Step Solution

Answer

The reference angle is \(\frac{\pi}{6}\) radians or 30 degrees

Step 1: Understand the value and position of the angle

Since the angle is negative, it is measured clockwise starting from the positive x-axis. Given angle is \(-\frac{5 \pi}{6}\). The radian measure \(\pi\) is equivalent to 180 degrees. Therefore, the angle measures \(-\frac{5 * 180}{6} = -150\) degrees clockwise from the positive x-axis.

Step 2: Sketch the angle

Draw a straight horizontal line representing the initial side on the positive x-axis. Measure an angle of 150 degrees clockwise from the horizontal line and draw a line at this angle. This line represents the terminal side of the angle. The angle \(-\frac{5 \pi}{6}\) is the angle between the horizontal line and this line.

Step 3: Calculate the reference angle

The reference angle is the smallest positive angle made with the x-axis. Since the angle is more than \(\frac{\pi}{2}\) radians (or 90 degrees), the reference angle will be \(\pi - |\text{given angle}|\), which will be \(\pi - |-\frac{5 \pi}{6}|\), which simplifies to \(\frac{\pi}{6}\) radians or equivalently 30 degrees.

Key Concepts

Angle Sketching

Angle sketching is a foundational skill in trigonometry. When sketching an angle, you start from the positive x-axis. This is your reference line. To sketch a given angle, you imagine the movement starting from this line.

If the angle is positive, you measure it in a counterclockwise direction. If it is negative, like \(-\frac{5 \pi}{6}\), you measure it clockwise. In practice, drawing a straight horizontal line helps mark the initial side. Then, based on the angle's value, move clockwise or counterclockwise to draw the terminal side.

• The initial side starts on the positive x-axis.
• For negative angles, move clockwise.
• For positive angles, move counterclockwise.

These sketches allow you to visually understand angle relationships and their positions on the coordinate plane.

Negative Angles

Understanding negative angles can be tricky at first. A negative angle indicates a clockwise rotation from the positive x-axis. In our case, \(-\frac{5 \pi}{6}\) means a rotation of 150 degrees clockwise.

Negative angles simply measure rotations in the opposite direction to positive angles.

• Negative means clockwise motion.
• Compare this with positive angles, which move counterclockwise.

This directionality should become second nature as you practice sketching angles, helping reinforce the concept.

Radian Measure

Radian measure is the standard unit of angular measurement in mathematics. One radian is the angle formed when the arc length is equal to the radius of the circle.

The given angle,\(-\frac{5 \pi}{6}\), can be converted between radians and degrees. Here, \(\pi\) radians are equivalent to 180 degrees. So, multiplying by \(\frac{180}{\pi}\) converts it to degrees:

• \(-\frac{5 \pi}{6} \times \frac{180}{\pi} = -150\) degrees

Understanding this conversion helps in switching between angles in different measurements for various applications.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of the coordinate plane. It is a crucial tool for visualizing angles, including their reference angles.

For angles like \(-\frac{5 \pi}{6}\), the unit circle aids in finding the reference angle. The reference angle is the smallest positive angle from the terminal side to the x-axis.

• Reference angle from \(-\frac{5 \pi}{6}\) is \(\frac{\pi}{6}\), or 30 degrees.

This understanding connects the angle's position to the circle's arc, essential for solving trigonometric problems.