Problem 76 - Precalculus : Building Concepts and Connections

Question

Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\csc \left(-\frac{5 \pi}{4}\right)$$

Step-by-Step Solution

Answer

Therefore, \( \csc(-\frac{5 \pi}{4}) = \sqrt{2} \).

Step 1: Determine the Trigonometric Function Value

The problem requires finding the cosecant function of \( \frac{5 \pi}{4} \). Thus, calculate the cosecant of \( \frac{5 \pi}{4} \). The values of trigonometric functions at special angles like \( \frac{\pi}{4} \), \( \frac{\pi}{2} \), \( \frac{3 \pi}{4} \), \( \pi \), etc. are known. And, since the cosecant is the reciprocal of the sine function, determining the sine of \( \frac{5 \pi}{4} \) simplifies the computation. Since \( \frac{5 \pi}{4} \) lies in the third quadrant where sine is negative, \( \sin(\frac{5 \pi}{4}) = -\sin(\frac{\pi}{4}) = -\frac{1}{\sqrt{2}} \). Therefore, \( \csc(\frac{5 \pi}{4}) = -\sqrt{2} \) as it is the reciprocal of sine.

Step 2: Apply the Negative-angle Identity

Now, apply the negative-angle identity for the cosecant function, which is \( \csc(-\theta) = -\csc(\theta) \). Substitute \( \theta = \frac{5 \pi}{4} \) into this identity. Hence, \( \csc(-\frac{5 \pi}{4}) = -\csc(\frac{5 \pi}{4}) \). Substitute the value from Step 1, \( \csc(\frac{5 \pi}{4}) = -\sqrt{2} \), into this equation. This yields \( \csc(-\frac{5 \pi}{4}) = -(-\sqrt{2}) \)

Key Concepts

Negative-Angle Identity

The negative-angle identity is a key concept in trigonometry that helps us understand how trigonometric functions behave when their angles are negative. In simple terms, the negative-angle identity provides a way to relate the function of a negative angle to a positive angle.

For instance, the negative-angle identity for the cosecant function is expressed as \( \csc(-\theta) = -\csc(\theta) \). This means that the cosecant of a negative angle is negative of the cosecant of the corresponding positive angle.

• This identity is particularly useful when evaluating trigonometric functions with negative angles, as it allows us to convert these into more familiar positive angles.
• By doing this, we can apply our knowledge of trigonometric functions at known angles.

Understanding this identity can simplify calculations and offers a method to work with negative angles more comfortably.

Cosecant Function

The cosecant function, symbolized as \( \csc(\theta) \), is one of the six main trigonometric functions. It is defined as the reciprocal of the sine function. This means:

\[ \csc(\theta) = \frac{1}{\sin(\theta)} \]

The cosecant function is rarely found on its own in practical scenarios but is rather important for theoretical understanding and solving trigonometric identities.

• The function is undefined whenever the sine of the angle is zero, since division by zero is not possible.
• As a reciprocal function, the cosecant will inherit the properties of the sine, such as the sign depending on the quadrant the angle is in.

Thus, if you know the sine of an angle \( \theta \), you can easily find its cosecant.

Reciprocal Functions

Reciprocal functions in trigonometry form a foundational part of understanding trigonometric identities. These functions are simply the reciprocals of the basic trigonometric functions: sine, cosine, and tangent.

• The cosecant function is the reciprocal of sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
• The secant function is the reciprocal of cosine: \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
• The cotangent function is the reciprocal of tangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

Understanding these reciprocal relationships is crucial when solving trigonometric equations and when simplifying or transforming expressions.

Reciprocal functions allow us to explore angles and their relationships in more depth, especially when one of the primary functions becomes zero, where their reciprocal would be undefined. These functions broaden the scope of trigonometry and provide alternative perspectives for analyzing trigonometric problems.