Problem 43 - Algebra and Trigonometry

Question

find the reference angle for each angle. $$ \frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

The reference angle for \( \frac{5\pi}{6} \) is \( \frac{\pi}{6} \)

Step 1: Conversion

Convert the radian measure to degrees, if necessary. We can skip this step as our problem is already in radians.

Step 2: Identify the quadrant

Identifying the quadrant where the angle moves is necessary to find the reference angle. The given angle \( \frac{5\pi}{6} \) is in the second quadrant.

Step 3: Find the Reference Angle

The reference angle in the second quadrant is calculated as \( \pi \) - the given angle. So, \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \) is the reference angle.

Key Concepts

Radian Measure

Radian measure is a way to express angles using the radius of a circle. Instead of degrees, radian measure uses the idea of the angle created by taking the radius of a circle and wrapping it along the circle's edge. This results in a unit called radians.
One of the key conversions is that 180 degrees is equivalent to \( \pi \) radians. Therefore, a full circle, which is 360 degrees, is equal to \( 2\pi \) radians. This conversion is crucial when working with trigonometric functions or solving problems involving circles, including finding reference angles.

• An angle of \( \pi/6 \) radians, for example, is exactly 30 degrees.
• \( \pi/2 \) radians are commonly known as 90 degrees.
• When dealing with radians, calculations often become more straightforward, particularly in higher-level mathematics.

Understanding radian measure is essential as it forms the foundation for many trigonometric calculations.

Angle Quadrants

Angling around the cartesian coordinate system, the "quadrants" are the four sections of the plane. Each quadrant has its characteristics and applications when analyzing angles. Knowing which quadrant an angle is located in helps determine its reference angle and aids trigonometric function calculations.
Angles are divided among the quadrants as follows:

• Quadrant I: 0 to \( \pi/2 \) radians (0 to 90 degrees)
• Quadrant II: \( \pi/2 \) to \( \pi \) radians (90 to 180 degrees)
• Quadrant III: \( \pi \) to \( 3\pi/2 \) radians (180 to 270 degrees)
• Quadrant IV: \( 3\pi/2 \) to \( 2\pi \) radians (270 to 360 degrees)

For example, the angle \( \frac{5\pi}{6} \) is located in the second quadrant since it falls between \( \pi/2 \) and \( \pi \), which is between 90 and 180 degrees. Recognizing the angle's quadrant is essential for determining the correct reference angle.

Trigonometric Calculations

Trigonometric calculations revolve around determining the sides and angles of triangles, using trigonometric functions like sine, cosine, and tangent. These calculations are fundamental in mathematics because they allow for the measurement and analysis of geometric shapes, especially circles.
When working with angles in the coordinate plane, the reference angle helps simplify these trigonometric calculations. A reference angle is the smallest angle formed by the terminal side of the given angle and the x-axis. It measures how far an angle is from the closest x-axis for quadrants I through IV.

• In Quadrant I, the reference angle is simply the angle itself.
• For Quadrant II, the reference angle is \( \pi \) minus the angle.
• In Quadrant III, it is the angle minus \( \pi \).
• For Quadrant IV, \( 2\pi \) minus the angle gives the reference angle.

For the problem of finding the reference angle of \( \frac{5\pi}{6} \), the angle belongs to the second quadrant. Therefore, the reference angle is computed as \( \pi - \frac{5\pi}{6} = \frac{\pi}{6} \). This reference angle is vital for solving trigonometric problems efficiently.