Question

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \csc 390^{\circ} $$

Short answer

\( \csc 390^{\circ} = 2 \)

Step-by-step solution

Determine the Periodicity of the Function

The cosecant function, \(\text{csc}\), is the reciprocal of the sine function, which has a period of \(360^{\circ}\). This means that \(\text{csc}\theta\) has the same period of \(360^{\circ}\).

Find the Equivalent Angle Within One Period

To find the equivalent angle for \(390^{\circ}\) within the interval \([0^{\circ}, 360^{\circ})\), subtract \(360^{\circ}\): \[ 390^{\circ} - 360^{\circ} = 30^{\circ} \]. Thus, \(\text{csc} 390^{\circ} = \text{csc} 30^{\circ}\).

Find the Cosecant of the Equivalent Angle

The angle \(30^{\circ}\) is a standard angle with a known sine value. \( \text{sin} 30^{\circ} = \frac{1}{2} \). Therefore, \(\text{csc} 30^{\circ} = \frac{1}{\text{sin} 30^{\circ}} = \frac{1}{\frac{1}{2}} = 2\).

Key concepts

cosecant function

The cosecant function, denoted as \( \text{csc} \theta \), is the reciprocal of the sine function. This means that \( \text{csc} \theta = \frac{1}{\text{sin} \theta} \). When working with trigonometric functions, it's crucial to understand their properties and behavior, especially when it comes to their reciprocal relationships.
Remember, since sine can never be zero (because division by zero is undefined), the cosecant function is undefined wherever sine is zero.
Key steps to finding the cosecant of an angle include:

• Understanding the relationship between sine and cosecant
• Using known sine values of standard angles
• Paying attention to the domain restrictions

For instance, at \( 30^{\text{°}} \), \( \text{sin} 30^{\text{°}} = \frac{1}{2} \), making \( \text{csc} 30^{\text{°}} = \frac{1}{\frac{1}{2}} = 2 \). Recognize the reciprocal nature and apply it to find exact values efficiently.

periodic functions

In trigonometry, periodic functions repeat their values in regular intervals. This is particularly essential when dealing with angles outside the standard range (0° to 360°). For any trigonometric function, the periodicity helps in simplifying expressions involving larger angles. For instance:
- The sine and cosine functions both have a period of \(360^{\text{°}} \) or \(2\text{π} \) radians.
- Therefore, \( \text{sin} ( \theta + 360^{\text{°}} ) = \text{sin} \theta \).
- Similarly, \( \text{csc} ( \theta + 360^{\text{°}} ) = \text{csc} \theta \).
In the given problem, knowing the periodicity helps us simplify \( \text{csc} 390^{\text{°}} \) to an equivalent angle within the basic interval:

• Subtract the period (360°) from 390°
• \( 390^{\text{°}} - 360^{\text{°}} = 30^{\text{°}} \)
• Thus, \( \text{csc} 390^{\text{°}} = \text{csc} 30^{\text{°}} \)

This periodic property significantly simplifies calculations, ensuring any angle’s trigonometric value falls within a familiar range.

standard angle

Standard angles are specific angles with known trigonometric values. Familiarizing yourself with them can make solving trigonometry problems much easier. Some common standard angles include 0°, 30°, 45°, 60°, and 90°. For these angles, the sine, cosine, and their reciprocal values are usually memorized:

• \( \text{sin} 30^{\text{°}} = \frac{1}{2}, \ \text{cos} 30^{\text{°}} = \frac{\text{√3}}{2} \)
• \( \text{csc} 30^{\text{°}} = 2, \ \text{sec} 30^{\text{°}} = \frac{2}{\text{√3}} \)
• and for other angles similarly

In the given problem, recognizing \(390^{\text{°}}\) relation to a standard angle (by periodicity) simplifies finding \( \text{csc} 30^{\text{°}} \) without much calculation. This is essential for quickly arriving at answers and seeing patterns in trigonometric problems.