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Chapter 5: Problem 12

Sketch the angles in standard position. $$-225^{\circ}$$

Short Answer

Expert verified
The sketch will look like a anticlockwise drawing from the positive x-axis going through all the quadrants, stopping at the second quadrant since it is \(-225^{\circ}\).

Step by step solution

01

Understand Standard Position

An angle is in standard position if it is drawn on an x-y plane with its vertex at the origin (0,0) and with the initial side on the positive x-axis. Rotations are usually counterclockwise for positive angles, and clockwise for negative angles.
02

Drawing 225 Degree Angle in Standard Position

Given the angle of \(-225^{\circ}\), we are asked to sketch it in standard position. A complete rotation is \(360^{\circ}\). The direction of rotation is clockwise as it is a negative angle. Starting from the positive x-axis, we proceed \(225^{\circ}\) in clockwise (negative direction).
03

Sketch the terminal side

Once having sketch the angle, we must draw the terminal side of \(-225^{\circ}\) to complete the angle in standard position. We would end up in the second quadrant as \(-225^{\circ}\) is \(135^{\circ}\) short of a complete \(360^{\circ}\) rotation in negative direction.

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Key Concepts

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

Standard Position
An angle is said to be in standard position when it is drawn in the coordinate plane with certain specific conditions. Here are the key points to remember about standard position:
  • The vertex of the angle is placed at the origin, which is \(0,0\) on the coordinate plane.
  • The initial side of the angle lies along the positive x-axis.
  • The angle measurement starts from the positive x-axis.
  • Positive angles result from counterclockwise rotation, while negative angles arise from clockwise rotation.
Understanding these aspects makes it easier to recognize and sketch angles correctly. The standard position ensures a consistent method of measuring and depicting angles in mathematics, allowing for uniformity in how angles are depicted and interpreted.
Negative Angles
Negative angles can sometimes be confusing, but are essential for full comprehension of angle measurement. Here's what you need to know:
  • Negative angles result from a clockwise rotation rather than the more typical counterclockwise one.
  • The starting point is still the positive x-axis, similarly to positive angles.
  • As the rotation is clockwise, the degrees are measured in the negative direction.
  • For example, an angle of \(-225^\circ\) involves moving 225 degrees in a clockwise direction from the positive x-axis.
Negative angles are not inherently different from positive angles, but they require attention to direction. Understanding how to measure and represent them is crucial for tasks involving angle calculations, like sketching angles in a standard coordinate plane.
Coordinate Plane Sketching
Sketching angles on the coordinate plane is a skill that builds upon understanding standard position and handling negative angles. Here are the steps to take when tasked with sketching an angle:
  • First, ensure the angle is in standard position with the vertex at (0,0) and one side on the positive x-axis.
  • Determine whether the angle is positive or negative to know the direction of rotation.
  • Perform the appropriate rotation:
    • For positive angles, rotate counterclockwise.
    • For negative angles, rotate clockwise.
  • Locate the terminal side of the angle after the rotation. For example, a \(-225^\circ\) angle lands in the second quadrant.
When sketching, it's often helpful to visualize a full 360-degree circle to find the angle's location quickly. Typically, understanding these basic concepts helps in creating accurate sketches on the coordinate plane.

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Most popular questions from this chapter

In this set of exercises, you will use inverse trigonometric functions to study real-world problems. Round all answers to four decimal places. A 15 -foot pole is to be stabilized by two wires of equal length, one on each side of the pole. One end of each wire is to be attached to the top of the pole; the other end is to be staked to the ground at an acute angle \(\theta\) with respect to the horizontal. Because of considerations, the ratio of the length of either wire to the height of the pole is to be no more than \(\frac{4}{3} .\) What is the limiting value of \(\theta\) in degrees? Is this limiting value a maximum value of \(\theta\) or a minimum value of \(\theta ?\) Explain.

Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\cot \left(-\frac{11 \pi}{6}\right)$$

Evaluate the given expressions to four decimal places with a calculator. $$\sec ^{-1}(-4.2)$$

Find the exact value of each expression without using a calculator. $$\tan \frac{\pi}{4} \sec \frac{\pi}{4}$$

Which is the larger angle, \(1^{\circ}\) or 1 radian? Explain.
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