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Chapter 13: Problem 42

Find a positive and a negative coterminal angle for the given angle. $$ -425^{\circ} $$

Short Answer

Expert verified
The positive coterminal angle is 295掳 and the negative coterminal angle is -785掳.

Step by step solution

01

Finding the positive coterminal angle

Starting with the given angle of -425掳, add 360掳 to get a positive coterminal angle. So we will calculate a = -425掳 + 360掳
02

Checking if the answer is positive

Check if the result from step 1 is positive. If it is not, keep adding 360掳 until a positive angle is obtained. Since -425掳 + 360掳 = -65掳, which is not positive, we need to add 360掳 again. So, calculate a = -65掳 + 360掳
03

Finding the negative coterminal angle

Starting with the given angle of -425掳, subtract 360掳 to get another negative coterminal angle. So b = -425掳 - 360掳
04

Checking if the result is negative

Check if the result from step 3 is negative. Since -425掳 - 360掳 = -785掳, which is already negative, there is no need to subtract 360掳 again. Thus the negative coterminal angle is -785掳.

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Key Concepts

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

Positive Coterminal Angle
When we talk about coterminal angles, we are essentially dealing with angles that share the same initial and terminal sides. These angles can be recognized by adding or subtracting multiples of 360掳.Finding a positive coterminal angle for a given angle, like -425掳, involves making sure the result is above zero.
  • Start with the given angle, -425掳.
  • Add 360掳 to get closer to a positive angle:\[ -425掳 + 360掳 = -65掳 \]
  • -65掳 is still negative, so we need to add another 360掳:\[ -65掳 + 360掳 = 295掳 \]
Thus, 295掳 is our positive coterminal angle for -425掳.It's always good to verify that the final angle is indeed positive, and keeps up with the rotations that wrap the angle back into a positive measure.
Negative Coterminal Angle
Negative coterminal angles are those which, while possibly several full rotations away, still line up the same as the initial angle in question.If you start with an angle like -425掳, one way to find a negative coterminal angle is by subtracting 360掳.
  • Begin with -425掳.
  • Subtract 360掳, yielding:\[ -425掳 - 360掳 = -785掳 \]
The good thing about this result is that it's already negative.There's no need for further adjustment unless you aim for a different rotation, but -785掳 already shares the same sides as -425掳.That's what makes them coterminal!
Angle Measurement
Understanding angle measurements is crucial because angles are fundamental in myriad areas, including geometry and trigonometry. Angles are commonly measured in degrees, which stems from dividing a circle into 360 degrees.
  • Coterminal angles demonstrate how the measure of an angle can extend beyond the typical 0掳 to 360掳 range.
  • The distinction among angles is established by the number and direction of 360掳 rotations applied to a starting angle.
  • Positive rotations are counterclockwise, while negative rotations are clockwise.
In different scenarios, knowing how to convert and find coterminal angles helps in simplifying calculations and understanding angular movements. Using coterminal angles efficiently aids in problem-solving, especially in cyclic and periodic functions.

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Most popular questions from this chapter

Writing Explain why each expression is undefined. $$ \cot 0^{\circ} $$

Find the exact value of each expression. If the expression is undefined, write undefined. $$ \sec 90^{\circ} $$

Indirect Measurement A communications tower has wires anchoring it to the ground. Each wire is attached to the tower at a height 20 \(\mathrm{ft}\) above the ground. The length \(y\) of the wire is modeled with the function \(y=20\) csc \(\theta,\) where \(\theta\) is the measure of the angle formed by the wire and the ground. a. Graph the function. b. Find the length of wire needed to form an angle of \(45^{\circ} .\) c. Find the length of wire needed to form an angle of \(60^{\circ} .\) d. Find the length of wire needed to form an angle of \(75^{\circ} .\)

a. Graph \(y-\tan x\) and \(y-\cot x\) on the same axes. b. State the domain, the range, and the asymptotes of each function. c. Writing Compare the two graphs. How are they alike? How are they different? d. Geometry The graph of the cotangent function can be reflected about a line to graph the tangent function. Name at least two lines that have this property.

a. What are the domain, range, and period of \(y=\csc x ?\) b. What is the relative minimum in the interval \(0 \leq x \leq \pi ?\) c. What is the relative maximum in the interval \(\pi \leq x \leq 2 \pi ?\)
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