Math Problem Statement

Find the reference angle for a rotation of start fraction, 13, pi, divided by, 7, end fraction 7 13π ​ .

Solution

To find the reference angle for the rotation of 13π7\frac{13\pi}{7}, follow these steps:

Step 1: Understand the concept of reference angles.

The reference angle is the smallest positive acute angle that the terminal side of the given angle makes with the x-axis. For angles measured in radians, the reference angle depends on which quadrant the terminal side lies in.

Step 2: Reduce the angle to an equivalent angle between 00 and 2π2\pi.

Given 13π7\frac{13\pi}{7}, the first thing is to reduce it within the interval [0,2π][0, 2\pi].

  • Since 2π=14π72\pi = \frac{14\pi}{7}, subtract 2π2\pi from 13π7\frac{13\pi}{7} to find the coterminal angle: 13π72π=13π714π7=π7\frac{13\pi}{7} - 2\pi = \frac{13\pi}{7} - \frac{14\pi}{7} = -\frac{\pi}{7}
  • A negative angle implies clockwise rotation, so add 2π2\pi to bring it into the positive interval: π7+2π=π7+14π7=13π7-\frac{\pi}{7} + 2\pi = -\frac{\pi}{7} + \frac{14\pi}{7} = \frac{13\pi}{7} Thus, the equivalent angle is π7\frac{\pi}{7}.

Step 3: Identify the quadrant and find the reference angle.

The reduced angle π7\frac{\pi}{7} is in the first quadrant, so it is already the reference angle.

Final Answer: The reference angle is π7\frac{\pi}{7}.

Would you like more details or clarifications? Here are some related questions:

  1. How do you convert between radians and degrees?
  2. What is the significance of reference angles in trigonometry?
  3. How do you find the reference angle for angles in the second, third, and fourth quadrants?
  4. How do coterminal angles help in determining reference angles?
  5. How can negative angles be interpreted geometrically?

Tip: To quickly reduce any angle larger than 2π2\pi (or 360360^\circ), subtract multiples of 2π2\pi (or 360360^\circ) to find a coterminal angle.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Radians
Reference Angle
Coterminal Angles

Formulas

Reference angle formula
Coterminal angles formula: θ - 2π or θ + 2π

Theorems

Reference Angle Theorem
Coterminal Angle Theorem

Suitable Grade Level

Grades 10-12