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Chapter 6: Problem 8

Find the reference angle for the given angle. (a) \(99^{\circ}\) (b) \(-199^{\circ}\) (c) \(359^{\circ}\)

Short Answer

Expert verified
(a) 81掳, (b) 19掳, (c) 1掳.

Step by step solution

01

Understanding Reference Angles

A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. This angle is always positive and measures less than or equal to 90掳. Reference angles are found by determining how far an angle is from the nearest x-axis, typically using absolute values and specific formulas depending on the quadrant.
02

Calculate Reference Angle for 99掳

Since 99掳 is in the second quadrant (greater than 90掳 but less than 180掳), the formula to find the reference angle is: \[180掳 - 99掳 = 81掳\]Thus, the reference angle for 99掳 is 81掳.
03

Adjust -199掳 to a Positive Angle

To handle negative angles, add 360掳 (since one full rotation is 360掳) until the angle is positive. \[-199掳 + 360掳 = 161掳\]Now consider 161掳.
04

Calculate Reference Angle for 161掳

161掳 is in the second quadrant. Use the formula \[180掳 - 161掳 = 19掳\]So the reference angle for -199掳 is 19掳.
05

Calculate Reference Angle for 359掳

Since 359掳 is in the fourth quadrant (greater than 270掳 but less than 360掳), the reference angle is found using the formula:\[360掳 - 359掳 = 1掳\]Therefore, the reference angle for 359掳 is 1掳.

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Key Concepts

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

Quadrants in Trigonometry
In trigonometry, the coordinate plane is divided into four sections known as quadrants. These quadrants aid in determining the characteristics and positions of angles. Each quadrant corresponds to a specific range of angle measures:
  • The first quadrant includes angles from 0掳 to 90掳. Here, all trigonometric functions like sine, cosine, and tangent are positive.
  • The second quadrant covers angles from 90掳 to 180掳. In this section, sine remains positive, while cosine and tangent are negative.
  • In the third quadrant, spanning from 180掳 to 270掳, both sine and cosine are negative, making tangent positive.
  • The fourth quadrant embraces angles from 270掳 to 360掳. Here, cosine stays positive, and sine and tangent are negative.
Identifying the quadrant of a given angle is a crucial step in finding its reference angle. It helps determine the correct formula to use based on whether you're subtracting from 180掳, adding to 180掳, or subtracting from 360掳. Understanding these quadrants aids in simplifying complex angle calculations and trigonometric analyses.
Calculating Angles
Calculating angles involves determining how far an angle's terminal side has rotated from the positive x-axis. This measurement can either be positive, moving counterclockwise, or negative, moving clockwise. Here are a few key points:
  • To calculate the reference angle, one must first determine the original angle's position in the coordinate plane or apply transformations for negative angles.
  • If an angle is negative, such as \(-199^{\circ}\), converting it to a positive angle by adding 360掳 can help in further calculations. In this case, adding 360掳 transforms \(-199^{\circ}\) to \(161^{\circ}\).
  • Reference angles are always measured from the terminal side of the given angle to the nearest x-axis, thereby ensuring they are always positive and less than 90掳.
By knowing these details, calculating the essential angle measures becomes straightforward. It's just about evaluating the given angle within its quadrant's context and applying the relevant formula.
Angle Measurement
Angles are measured in degrees, a unit of measurement that breaks a circle into 360 equally spaced segments. This system allows us to describe the size of an angle by how much it leans from the baseline. Understanding the basics of angle measurement is essential for solving problems involving reference angles.
  • Degrees are marked by the symbol 掳 and organization within a circle can be navigated using this unit.
  • A full rotation encompasses 360掳, while a half rotation accounts for 180掳.
  • Reference angles are those acute angles that serve as a reference to their respective positions on the unit circle by projecting to the nearest x-axis.
When tackling trigonometry problems, it is vital to measure angles accurately to apply proper methods for finding reference angles and solving other trigonometric challenges efficiently. Knowing that \(180^{\circ}\) and \(360^{\circ}\) represent pivotal points on the unit circle streamlines the analysis and comprehension of different angular measures.

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Most popular questions from this chapter

Distance Across a River To find the distance across a river, a surveyor chooses points \(A\) and \(B\), which are 200 ft apart on one side of the river (see the figure). She then chooses a reference point \(C\) on the opposite side of the river and finds that \(\angle B A C=82^{\circ}\) and \(\angle A B C \approx 52^{\circ} .\) Approximate the distance from \(A\) to \(C\). (Image can't copy)

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Tracking a Satellite The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations \(A\) and \(B,\) which are 50 mi apart. When the satellite is on one side of the two stations, the angles of elevation at \(A\) and \(B\) are measured to be \(87.0^{\circ}\) and \(84.2^{\circ},\) respectively. (a) How far is the satellite from station \(A ?\) (b) How high is the satellite above the ground? (Image can't copy)
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