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Chapter 7: Problem 45

Find the reference angle of each angle. $$ 320^{\circ} $$

Short Answer

Expert verified
The reference angle is 40掳.

Step by step solution

01

Identify the Quadrant

First, determine in which quadrant the given angle lies. The angle provided is 320掳. Since it is between 270掳 and 360掳, it lies in the fourth quadrant.
02

Use the Reference Angle Formula

For an angle in the fourth quadrant, the reference angle can be found using the formula: \[\text{Reference angle} = 360^{\text{o}} - \theta\] where \theta\ is the given angle.
03

Substitute and Calculate

Substitute the given angle into the reference angle formula: \[ \text{Reference angle} = 360^{\text{o}} - 320^{\text{o}} = 40^{\text{o}} \]

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Key Concepts

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

quadrants
In trigonometry, the coordinate plane is divided into four parts called quadrants. Each quadrant houses angles and gives insight into their properties. Understanding quadrants makes it easier to grasp angle behaviors.
  • First Quadrant: 0掳 to 90掳
  • Second Quadrant: 90掳 to 180掳
  • Third Quadrant: 180掳 to 270掳
  • Fourth Quadrant: 270掳 to 360掳
Knowing these can help you determine the reference angle, which is the smallest angle formed with the x-axis. For example, our angle of 320掳 falls into the fourth quadrant since it lies between 270掳 and 360掳.
angle measurement
An angle's measurement can be determined in degrees. Degrees are a way to measure how far you鈥檝e rotated from one point to another. In a circle, which is 360掳, we can split this into four right angles, each being 90掳. To visualize, picture a full circle divided into four equal parts. Each part represents one of the quadrants. Using degrees, we can pinpoint exactly where an angle is on a circle. By knowing the degrees, you can identify the quadrant it belongs to鈥攅ssential in problems like finding reference angles.
trigonometry formulas
Trigonometry relies on specific formulas to solve angles and sides of triangles. For finding reference angles, different quadrants use distinct formulas. Here's a brief overview:
  • First Quadrant: Reference angle = 胃
  • Second Quadrant: Reference angle = 180掳 - 胃
  • Third Quadrant: Reference angle = 胃 - 180掳
  • Fourth Quadrant: Reference angle = 360掳 - 胃
The formula for our 320掳 angle, lying in the fourth quadrant, becomes: \[Reference \, angle = 360^{\circ} - 320^{\circ} = 40^{\circ}\] Using these formulas helps simplify otherwise complex angle calculations.
degrees
Degrees are a crucial unit of measurement in trigonometry. They help us communicate and understand angles precisely. One complete rotation around a circle is 360掳, broken down into smaller parts.
  • 1掳 = 1/360 of a full rotation
  • 90掳 = right angle
  • 180掳 = straight line
In trigonometry, it鈥檚 common to also consider angles beyond 360掳, by effectively adding full circles. Degrees help us break down an angle to its smallest component, making it easier to find key values like reference angles. For example, the angle 320掳 can be referenced by subtracting from 360掳, resulting in a much smaller, easier to understand, 40掳 reference angle.

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Most popular questions from this chapter

Given \(\cot \theta=2,\) use trigonometric identities to find the exact value of (a) \(\tan \theta\) (b) \(\csc ^{2} \theta\) (c) \(\tan \left(\frac{\pi}{2}-\theta\right)\) (d) \(\sec ^{2} \theta\)

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