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Chapter 6: Problem 30

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are co terminal with the given angle. $$135^{\circ}$$

Short Answer

Expert verified
Positive co-terminal angles: 495掳, 855掳. Negative co-terminal angles: -225掳, -585掳.

Step by step solution

01

Understand Co-terminal Angles

Co-terminal angles are angles that share the same initial and terminal sides but can represent multiple full rotations. To find angles that are co-terminal with a given angle, you can add or subtract multiples of 360掳 from the given angle.
02

Find Positive Co-terminal Angles

Starting with the angle 135掳, add 360掳 to find the next positive co-terminal angle: \[135掳 + 360掳 = 495掳.\]Then, to find another positive co-terminal angle, add 360掳 again: \[495掳 + 360掳 = 855掳.\] So, 495掳 and 855掳 are positive co-terminal angles with 135掳.
03

Find Negative Co-terminal Angles

To find negative co-terminal angles, subtract 360掳 from the original angle:\[135掳 - 360掳 = -225掳.\]Subtract 360掳 again to find another negative co-terminal angle: \[-225掳 - 360掳 = -585掳.\] Thus, -225掳 and -585掳 are negative co-terminal angles with 135掳.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angles
Angles are a fundamental concept in trigonometry used to describe rotation or the amount of turn between two rays that have a common endpoint. This common endpoint is known as the vertex of the angle. There are several types of angles based on their measures:
  • **Acute angles:** These are angles less than 90掳.
  • **Right angles:** These are exactly 90掳.
  • **Obtuse angles:** These angles are more than 90掳 but less than 180掳.
  • **Straight angles:** These are exactly 180掳.
In the context of rotation or circular motion, angles can extend beyond the traditional 0掳 to 360掳 range via full rotations. For example, a 450掳 angle means one full rotation (360掳) plus 90掳. This idea is particularly important when discussing co-terminal angles, which are a key concept in understanding circular movements and periodic functions in trigonometry.
Co-terminal Angles
Co-terminal angles are angles that end at the same position after possibly multiple rotations. Imagine rotating a wheel multiple times -- no matter how many times it spins, its final position is what determines the angle relative to a starting line. To find co-terminal angles:
  • **Add or Subtract 360掳:** Start with a given angle and either add or subtract multiples of 360掳, to find angles that are co-terminal.
  • **Positive and Negative:** You can find positive and negative angles that are co-terminal with the original angle, offering flexibility in calculations and understanding.
For instance, the step-by-step solution calculated the positive co-terminal angles 495掳 and 855掳 by adding 360掳 repeatedly to the angle 135掳. It also found negative co-terminal angles -225掳 and -585掳 by subtracting 360掳 multiple times. This demonstrates how angles can traverse a full circle in both directions, yet arrive at the same terminal point.
Standard Position
The concept of "standard position" is used to give a consistent starting point for measuring angles. An angle in standard position has its vertex at the origin of a coordinate system, its initial side along the positive x-axis. Here's why the standard position is important:
  • **Consistency:** By establishing a fixed starting line, mathematicians and scientists can uniformly describe the position of angles, making it easier to communicate and understand rotational data.
  • **Graphical Illustration:** Angles in standard position can easily be plotted on a Cartesian plane, making visuals clearer and aiding in understanding spatial relationships.
For instance, the angle 135掳 in standard position will have its vertex at the origin, with the initial side on the positive x-axis and final side typically in the second quadrant. This setup helps in calculating co-terminal angles and observing how they visually lay out in a full rotation. Standard position acts as an anchor for interpreting angles in a two-dimensional space.

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