Problem 30 - Algebra 2 and Trigonometry

Question

In \(26-33 :\) a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. $$ \csc \frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

\(\csc \frac{5\pi}{6} = 2\).

Step 1: Understanding Cofunction Identities

Cofunction identities are trigonometric identities that relate the trigonometric functions of complementary angles. The cofunction identity for cosecant (csc) is related to the secant (sec) of the complementary angle: \[\csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right)\] Using this identity, we will rewrite the given cosecant function.

Step 2: Applying the Cofunction Identity

The given function is \(\csc \frac{5 \pi}{6}\). By applying the cofunction identity:\[\csc \frac{5\pi}{6} = \sec\left(\frac{\pi}{2} - \frac{5\pi}{6}\right)\]We need to compute \(\frac{\pi}{2} - \frac{5\pi}{6}\).

Step 3: Computing the Complementary Angle

Calculate the difference:\[\frac{\pi}{2} - \frac{5\pi}{6} \]Find a common denominator, which will be 6, to perform the subtraction:\[\frac{3\pi}{6} - \frac{5\pi}{6} = -\frac{2\pi}{6} = -\frac{\pi}{3}\]Now substitute this result back:\[\csc \frac{5\pi}{6} = \sec\left(-\frac{\pi}{3}\right)\]

Step 4: Simplifying the Cofunction

The secant function is even, meaning \(\sec(-x) = \sec(x)\). Therefore, \[\sec\left(-\frac{\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right)\] Next, find the exact value of \(\sec\left(\frac{\pi}{3}\right)\).

Step 5: Finding the Exact Trigonometric Value

The exact value of \(\sec\left(\frac{\pi}{3}\right)\) can be found from the unit circle or a trigonometric table. We know that:\[\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\]Thus:\[\sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2\]

Step 6: Conclusion

We have successfully rewritten the function in terms of its cofunction and calculated its exact value. Thus, \(\csc \frac{5\pi}{6} = 2\).

Key Concepts

Cosecant Function

The cosecant function, noted as \( \csc \theta \), is one of the fundamental trigonometric functions and is the reciprocal of the sine function. This means:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• It is undefined for angles where \( \sin \theta = 0 \)

This function is often used alongside others like secant and tangent to solve problems in trigonometry.
This particular exercise involves transforming the cosecant function into its cofunction using the identity related to secant. Understanding this relationship helps simplify complicated trigonometric expressions and solve equations more easily.

Secant Function

The secant function, denoted as \( \sec \theta \), is another key trigonometric function. It is defined as the reciprocal of the cosine function:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• It is undefined for angles where \( \cos \theta = 0 \)

This exercise particularly relies on understanding how to manipulate the secant function as part of its cofunction identity with cosecant.
Because the secant function is even, where \( \sec(-x) = \sec(x) \), it allows for broader applications in solving trigonometric equations across different angle measurements.

Complementary Angles

Complementary angles are two angles whose sum equals \( \frac{\pi}{2} \) radians, or 90 degrees. In trigonometry, understanding complementary angles is crucial for using various identities, such as cofunction identities.
Cofunction identities state that each trigonometric function of an angle is equal to the cofunction of its complementary angle. For example:

• \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \)
• \( \sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \)

These relationships form the basis for transforming trigonometric expressions, as seen in the provided solution where cosecant was expressed in terms of secant.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variable. These identities are essential tools in simplifying and solving trigonometric equations.
This exercise demonstrates the use of a specific type of trigonometric identity known as a cofunction identity. Cofunction identities are useful because they relate trigonometric functions to their counterparts of complementary angles. For example:

• \( \csc(\theta) = \sec\left(\frac{\pi}{2} - \theta\right) \)
• \( \tan(\theta) = \cot\left(\frac{\pi}{2} - \theta\right) \)

Using these identities can simplify complex trigonometric expressions and make calculations, like finding the exact value of function values, much more straightforward. Understanding and utilizing them effectively is a crucial skill in trigonometry.