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Magazine Chapter 2: Problem 52
Evaluate the following expressions exactly by using a reference angle. $$ \csc \left(-240^{\circ}\right) $$
Short Answer
Expert verified
\(\csc(-240^{\circ}) = \frac{2\sqrt{3}}{3}\).
Step by step solution
01
Identify the Angle in Standard Position
The given angle is \(-240^{\circ}\). First, find its equivalent angle in the standard position. To do this, add \(360^{\circ}\) until the angle is positive. Thus, \(-240^{\circ} + 360^{\circ} = 120^{\circ}\). Therefore, \(120^{\circ}\) is the angle in standard position.
02
Find the Reference Angle
The reference angle is the acute angle formed by the terminal side of \(120^{\circ}\) and the nearest x-axis. Since \(120^{\circ}\) is in the second quadrant, the reference angle \(\theta_{ref}\) is \(180^{\circ} - 120^{\circ} = 60^{\circ}\).
03
Determine the Cosecant of the Reference Angle
The cosecant function is the reciprocal of the sine function, and it is positive in both the first and the second quadrant. Using a reference angle of \(60^{\circ}\), we know that \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\). Therefore, \(\csc(60^{\circ}) = \frac{1}{\sin(60^{\circ})} = \frac{2}{\sqrt{3}}\).
04
Adjust for the Negative Angle
Since the angle given is \(-240^{\circ}\), and sine is positive in the second quadrant, \(\csc(-240^{\circ})\) is the same as \(\csc(120^{\circ})\), which equals \(\frac{2}{\sqrt{3}}\). Typically, \( \frac{2}{\sqrt{3}} \) is rationalized to \( \frac{2\sqrt{3}}{3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Reference Angle
To solve trigonometric expressions, understanding the concept of a reference angle is crucial. A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always positive and typically falls between 0掳 and 90掳.
In many problems, especially those involving angles greater than 90掳 or negative angles, the reference angle helps to determine the trigonometric functions of the original angle. To find a reference angle:
In many problems, especially those involving angles greater than 90掳 or negative angles, the reference angle helps to determine the trigonometric functions of the original angle. To find a reference angle:
- If the angle is in the first quadrant, the reference angle is the angle itself.
- For the second quadrant, subtract the angle from 180掳.
- In the third quadrant, subtract 180掳 from the angle.
- In the fourth quadrant, subtract the angle from 360掳.
Cosecant
The cosecant function is one of the six fundamental trigonometric functions. It is represented as \( ext{csc} \) and is the reciprocal of the sine function. This means that for any angle \( \theta \), the relationship is \( ext{csc}(\theta) = \frac{1}{ ext{sin}(\theta)} \).
Since the sine of an angle corresponds to the ratio of the opposite side to the hypotenuse in a right triangle, cosecant flips that ratio:
Since the sine of an angle corresponds to the ratio of the opposite side to the hypotenuse in a right triangle, cosecant flips that ratio:
- Cosecant is undefined for angles where sine is zero, such as 0掳 and 180掳.
- The sign of cosecant depends on the quadrant, matching the sign of sine. It is positive in the first and second quadrants.
Standard Position
An angle's standard position is a fundamental concept in trigonometry. An angle is said to be in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis.
This framework helps standardize calculations and interpretations of angles when extending beyond the simple range of 0掳 to 360掳. When a given angle is negative or greater than 360掳, adjustments are made by adding or subtracting multiples of 360掳 to find a corresponding angle within this range. This ensures that the angle is accurately represented for trigonometric evaluations.
For example, the angle \(-240掳\) is adjusted to \(120掳\) by adding \(360掳\).
This framework helps standardize calculations and interpretations of angles when extending beyond the simple range of 0掳 to 360掳. When a given angle is negative or greater than 360掳, adjustments are made by adding or subtracting multiples of 360掳 to find a corresponding angle within this range. This ensures that the angle is accurately represented for trigonometric evaluations.
For example, the angle \(-240掳\) is adjusted to \(120掳\) by adding \(360掳\).
Reciprocal Trigonometric Functions
In trigonometry, the reciprocal trigonometric functions are pivotal but often less directly referred to compared to their sine, cosine, and tangent counterparts. These include:
- Cosecant (csc), the reciprocal of sine: \( ext{csc}(\theta) = \frac{1}{ ext{sin}(\theta)} \).
- Secant (sec), the reciprocal of cosine: \( ext{sec}(\theta) = \frac{1}{ ext{cos}(\theta)} \).
- Cotangent (cot), the reciprocal of tangent: \( ext{cot}(\theta) = \frac{1}{ ext{tan}(\theta)} \).
