Problem 52 - Precalculus Essentials

Question

Express the exact value of each function as a single fraction. Do not use a calculator. $$\text { If } \theta \text { is an acute angle and } \cos \theta=\frac{1}{3}, \text { find } \csc \left(\frac{\pi}{2}-\theta\right)$$.

Step-by-Step Solution

Answer

The exact value of the function \( \csc \left(\frac{\pi}{2}-\theta\right) \) is 3.

Step 1: Identify the expression

The target expression is \( \csc \left(\frac{\pi}{2}-\theta\right) \), where \( \cos \theta = \frac{1}{3} \).

Step 2: Apply trigonometric identity

This step involves recognizing that \( \csc \left(\frac{\pi}{2}-\theta\right) \) is equivalent to \( \sec \theta \), due to the co-function identity of trigonometric functions which states: \( \csc \left(\frac{\pi}{2}-\theta\right) = \sec \theta \). This is a standard trigonometric result which simplifies the problem.

Step 3: Evaluate \( \sec \theta \)

The secant of an angle is the reciprocal of the cosine of the angle. Since \( \cos \theta = \frac{1}{3} \), it follows that \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{1}{3}} = 3 \).

Key Concepts

Co-function Identity

Trigonometric functions have fascinating relationships among each other. One such relationship is the co-function identity. Co-functions are pairs of trigonometric functions where the function of an angle is equal to the co-function of its complementary angle. A complementary angle is what remains when you subtract your angle from 90° (or \( \frac{\pi}{2} \) radians).

The co-function identity used in our exercise can be expressed as follows:

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

In the given problem, we’re asked to find \( \csc \left(\frac{\pi}{2}-\theta\right) \). According to the co-function identity, this equals \( \sec \theta \). Knowing these identities helps us to transition smoothly between different trigonometric expressions.

Secant

The secant function is one of the six fundamental trigonometric functions. It is closely related to the cosine function. Specifically, the secant of an angle is the reciprocal of the cosine of that angle. So, if you know \( \cos \theta \), it's quite easy to find \( \sec \theta \)!

Mathematically, this relationship is expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

In our exercise, we were given that \( \cos \theta = \frac{1}{3} \). To find \( \sec \theta \), we simply took the reciprocal:

• \( \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\left(\frac{1}{3}\right)} = 3 \)

Understanding secant as the reciprocal of cosine aids significantly in solving many trigonometric problems quickly and efficiently.

Reciprocal Trigonometric Functions

Trigonometry is not just about sine, cosine, and tangent. It also includes their reciprocal functions: cosecant, secant, and cotangent. Reciprocal functions are inverse relationships in terms of multiplication with the base trigonometric functions to equal one. These relationships can be handy when dealing with fractions or simplifying expressions.

The reciprocal relationships are as follows:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

For example, in our exercise, we used the reciprocal identity of secant to solve for \( \sec \theta \) because \( \cos \theta \) was provided. In essence:

• Knowing one trigonometric function allows you to easily find its reciprocal.
• These identities are key tools for simplifying complex trigonometric expressions.

Having a firm grip on these reciprocal relationships unlocks plenty of shortcuts in trigonometry problems!