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Chapter 1: Problem 38

Sketch each angle in standard position. (a) \(-750^{\circ}\) (b) \(-600^{\circ}\)

Short Answer

Expert verified
The coterminal positive equivalents for the angles \(-750^{\circ}\) and \(-600^{\circ}\) are \(-30^{\circ}\) and \(120^{\circ}\) respectively. Sketching these angles in standard position leads to the desired result of mapping \(-750^{\circ}\) (clockwise 30 degrees from the positive x-axis) and \(-600^{\circ}\) (counterclockwise 120 degrees from the positive x-axis).

Step by step solution

01

Normalize Negative Angle -750 Degrees

Angle \(-750^{\circ}\) is the angle that needs to be sketched first. Since it's negative, it means the rotation is clockwise. But we can find its positive coterminal angle by adding 360 degrees (or multiples of it) until we get a positive angle less than 360. So \( -750 + 2 \cdot 360 = -30^{\circ}\), which represents the same rotation in standard position.
02

Step 2. Sketch angle -750 degrees

Now to sketch angle \(-30^{\circ}\) (the coterminal angle to \(-750^{\circ}\)), draw a set of x and y axes. From the positive x-axis, move clockwise around the origin by 30 degrees and draw a line from the origin at this angle. Label this line as \(-750^{\circ}\). The result is your sketch for angle \(-750^{\circ}\) in standard position.
03

Normalize Negative Angle -600 Degrees

Angle \(-600^{\circ}\) is the second angle that requires sketching. Again, it is negative meaning the rotation is clockwise. Find its positive coterminal angle by adding 360 degrees (or multiples of it) until reaching a positive angle less than 360. So \( -600 + 2 \cdot 360 = 120^{\circ}\), which represents the same rotation in standard position.
04

Sketch angle -600 degrees

To sketch angle \(120^{\circ}\) (the coterminal angle to \(-600^{\circ}\)), draw a set of x and y axes if not already there from previous step. From the positive x-axis, move counterclockwise around the origin by 120 degrees and draw a line from the origin at this angle. Label this line as \(-600^{\circ}\). The result is your sketch for angle \(-600^{\circ}\) in standard position.

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Key Concepts

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

Coterminal Angles
Coterminal angles are angles that share the same terminal side when drawn in standard position on a coordinate plane. You can find coterminal angles by adding or subtracting multiples of 360 degrees to a given angle.
For example, both \(-750^{\circ}\) and \(-30^{\circ}\) are coterminal, as adding 720 degrees to \(-750^{\circ}\) results in \(-30^{\circ}\).
  • Coterminal angles allow us to simplify angle measures.
  • If you have a negative angle, find a positive coterminal angle by adding 360 degrees until you achieve a positive result.
This concept helps in creating easier representations for sketching and analyzing angles.
Standard Position
In trigonometry, an angle is said to be in standard position if its vertex is at the origin of a coordinate system, and its initial side lies along the positive x-axis. The terminal side depends on the angle's measure.
To visualize this:
  • Draw the initial side along the positive x-axis.
  • Rotate counterclockwise for positive angles, and clockwise for negative angles.
This method ensures consistency and clarity when dealing with angles, making it simpler to understand relationships between them.
Negative Angles
Negative angles indicate a clockwise rotation from the positive x-axis. They can be confusing, but remember:
  • Negative means rotate to the opposite direction of positive.
  • Negative angles can be converted to positive angles by finding their coterminal equivalents.
For example, when given \(-600^{\circ}\), it can be translated to a positive \(120^{\circ}\), thanks to coterminal angles. Understanding both positive and negative angles helps bridge rotational directions and provides a full picture of rotational motion.

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Most popular questions from this chapter

In Exercises 109-112, sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side. Then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \cos \theta=\frac{5}{6} $$

In Exercises 59-68, write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 7.) $$ \sin (\arccos x) $$

In Exercises 1-16, evaluate the expression without using a calculator. $$ \tan ^{-1}\left(-\frac{\sqrt{3}}{3}\right) $$

In Exercises 83-88, use a graphing utility to graph the function. $$ f(x)=\pi \arcsin (4 x) $$

In Exercises 89 and 90, write the function in terms of the sine function by using the identity $$ A \cos \omega t+B \sin \omega t=\sqrt{A^{2}+B^{2}} \sin \left(\omega t+\arctan \frac{A}{B}\right) $$ Use a graphing utility to graph both forms of the function. What does the graph imply? $$ f(t)=4 \cos \pi t+3 \sin \pi t $$
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