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Chapter 2: Problem 25

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

Short Answer

Expert verified
The angle -225掳 is coterminal with 135掳 and is in the second quadrant.

Step by step solution

01

Identify the Number of Full Rotations

First, determine how many full rotations of 360 degrees are in -225 degrees. Dividing -225 by 360 gives approximately -0.625, indicating that the angle involves a negative rotation of less than one full circle.
02

Determine Coterminal Angle

Find a coterminal angle by adding full rotations of 360 degrees to -225 degrees until you have an angle between 0 and 360 degrees. By adding one full rotation (360 degrees): \[-225^{\circ} + 360^{\circ} = 135^{\circ}\].This means -225 degrees is coterminal with 135 degrees.
03

Visualize the Angle

Locate 135 degrees in standard position on a coordinate plane. It starts from the positive x-axis moving counterclockwise, ending in the second quadrant. Since -225 degrees and 135 degrees are coterminal, sketch 135 degrees as the representation of -225 degrees.

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Key Concepts

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

Standard Position Angles
When we talk about standard position angles, we're referring to a specific way of placing an angle on a coordinate plane. This is crucial in understanding the way angles are measured in trigonometry.
  • The vertex of the angle is placed at the origin of the coordinate system.
  • The initial side of the angle lies along the positive x-axis.
  • For a positive angle, the terminal side - where the angle measurement ends - is rotated counterclockwise.
Understanding standard position angles helps in predicting where the terminal side will be, based on the angle measure. So even for a negative angle like -225掳, once you find its coterminal angle (135掳), it's easier to visualize because 135掳 starts at the positive x-axis and moves counterclockwise into the second quadrant.
Negative Angle Rotation
Negative angles can seem a bit tricky at first. Let's break it down so you can easily grasp the concept of negative angle rotation.

When we have a negative angle, it simply means we rotate clockwise from the positive x-axis instead of counterclockwise.
  • A negative rotation such as -225掳 means you go clockwise 225 degrees.
  • However, we often convert it to a positive angle by adding 360 degrees to find an equivalent angle.
For example, -225掳 plus one full circle (360掳) equals 135掳. This means -225掳 ends up in the same place on the coordinate plane as 135掳. Converting negative angles to positive ones makes sketching them in standard position easier.
Coordinate Plane
The coordinate plane is a visual tool that makes sketching and understanding angles much clearer.

It consists of two axes:
  • The horizontal axis is the "x-axis."
  • The vertical axis is the "y-axis."
The intersection of these two axes is the origin, where the vertex of our angle in standard position sits. When drawing an angle:
  • Start at the origin.
  • Move along the positive x-axis for the initial side.
  • Rotate to reach the terminal side depending on the angle's measurement, moving counterclockwise for positive angles and clockwise for negative ones.
Understanding the coordinate plane is key in visualizing and sketching angles - it allows you to see exactly which quadrant an angle will end up in based on how much you've rotated from the initial side.

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Most popular questions from this chapter

Write in terms of \(\sin \theta: \frac{\cos \theta}{\tan \theta(1-\sin \theta)}\).

Use a quotient identity to find the function value indicated. Rationalize denominators if necessary. If \(\sin \theta=-\frac{1}{2}\) and \(\cos \theta=\frac{\sqrt{3}}{2}\), find \(\cot \theta\).

In Exercises 65 and 66, explain the mistake that is made. Find the angle with smallest positive measure that is coterminal with the angle with measure \(-45^{\circ}\). Assume that both angles are in standard position. Solution: Coterminal angles are supplementary angles. \(-45^{\circ}+\alpha=180^{\circ}\) Add \(45^{\circ}\) to both sides. \(\alpha=225^{\circ}\) This is incorrect. What mistake was made?

If \(\theta=165^{\circ}\), find the measure of the reference angle. What is the physical meaning of the reference angle?

If light rays hit the water's surface at an angle \(30^{\circ}\) from the perpendicular and are refracted to an angle of \(22^{\circ}\) from the perpendicular, then what is the refraction index for water?
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