Problem 30 - Algebra and Trigonometry

Question

Find the exact value of the expression. $$ \csc \left(\cos ^{-1} \frac{7}{25}\right) $$

Step-by-Step Solution

Answer

The exact value is \( \frac{25}{24} \).

Step 1: Understand the Inverse Cosine Function

We are given \[ \csc \left(\cos^{-1} \frac{7}{25}\right) \]This implies that we need to evaluate the cosecant of an angle whose cosine is \( \frac{7}{25} \). Let's denote this angle by \( \theta \). Therefore, \( \cos \theta = \frac{7}{25} \).

Step 2: Use the Pythagorean Identity

To find the sine of \( \theta \), we use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \]Substitute \( \cos \theta = \frac{7}{25} \) into the equation:\[ \sin^2 \theta + \left( \frac{7}{25} \right)^2 = 1 \]This simplifies to:\[ \sin^2 \theta + \frac{49}{625} = 1 \]\[ \sin^2 \theta = 1 - \frac{49}{625} \]

Step 3: Calculate \( \sin \theta \)

Subtract \( \frac{49}{625} \) from 1:\[ 1 - \frac{49}{625} = \frac{625 - 49}{625} = \frac{576}{625} \]Thus, we have:\[ \sin^2 \theta = \frac{576}{625} \]Taking the square root (considering the principal value, assuming \( \theta \) is in the range of \( \cos^{-1} \)), we obtain:\[ \sin \theta = \frac{24}{25} \]

Step 4: Find \( \csc \theta \)

The cosecant of an angle is the reciprocal of its sine. Thus:\[ \csc \theta = \frac{1}{\sin \theta} \]Substitute \( \sin \theta = \frac{24}{25} \):\[ \csc \theta = \frac{1}{\frac{24}{25}} = \frac{25}{24} \]

Step 5: Conclusion

Therefore, the exact value of the given expression is:\[ \csc \left(\cos^{-1} \frac{7}{25}\right) = \frac{25}{24} \]

Key Concepts

Cosecant Function

The cosecant function can be understood as one of the six primary trigonometric functions, closely related to sine. In fact, it is the reciprocal of the sine function. To find the cosecant of an angle \(\theta\), use the formula:

• \( \csc \theta = \frac{1}{\sin \theta} \)

It's important to note that cosecant is defined for all angles where sine is not zero since division by zero is undefined. Whenever you encounter a problem involving cosecant, the strategy involves first finding the sine of the given angle and then taking its reciprocal.
In the context of the given problem, the need arises to calculate \( \csc \theta \), where \( \theta \) is derived from the expression \( \csc(\cos^{-1}\frac{7}{25}) \). By first determining \( \sin \theta \) using trigonometric identities, we can easily find \( \csc \theta \) by taking the reciprocal of the sine value. This basic grasp of cosecant as a reciprocal process streamlines the problem-solving approach.

Inverse Cosine Function

The inverse cosine function, denoted as \( \cos^{-1} x \) or sometimes \( \text{arccos} x \), gives us the angle \( \theta \) whose cosine is \( x \). In other words, if \( \cos \theta = x \), then \( \cos^{-1} x = \theta \). This function is critical in the realm of trigonometry for finding angles when their cosine is known.
One of the main characteristics of \( \cos^{-1} x \) is its range, which is between 0 and \( \pi \) radians (or 0° to 180° in degrees). This range ensures that for every valid input, a unique output (angle) is returned.
In solving the problem \( \csc(\cos^{-1}\frac{7}{25}) \), \( \cos^{-1} \frac{7}{25} \) leads us to find \( \theta \) where \( \cos \theta = \frac{7}{25} \). With \( \theta \) identified, the process of determining the corresponding sine (and consequently the cosecant) becomes feasible.

Pythagorean Identity

The Pythagorean identity is one of the foundational relationships in trigonometry, derived from the Pythagorean theorem. It links the square of sine and cosine functions of an angle. The identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This formula is utilized to determine one trigonometric function if the other is known. If you have \( \cos \theta \), you can find \( \sin \theta \) using the identity, and vice versa.
In the textbook exercise \( \csc(\cos^{-1}\frac{7}{25}) \), since \( \cos \theta = \frac{7}{25} \), the Pythagorean identity helps compute \( \sin \theta \). By substituting \( \cos \theta \) into the identity, \( \sin^2 \theta \) can be calculated leading to \( \sin \theta = \frac{24}{25} \).
This powerful identity not only assists in solving such problems but also ensures that solutions are aligned with fundamental trigonometric truths, making it an essential tool in any trigonometry toolkit.