Problem 46 - Precalculus: Functions and Graphs

Question

Use fundamental Identities to write the first expression in terms of the second, for any acute angle \(\boldsymbol{\theta}\). $$\csc \theta, \cos \theta$$

Step-by-Step Solution

Answer

\( \csc \theta = \frac{1}{\sqrt{1 - \cos^2 \theta}} \)

Step 1: Recall the Cosecant Identity

The cosecant function is the reciprocal of the sine function. The identity relating cosecant to sine is given by: \[ \csc \theta = \frac{1}{\sin \theta} \]

Step 2: Relate Sine and Cosine

We know that for any angle \( \theta \), the Pythagorean identity is: \[ \sin^2 \theta + \cos^2 \theta = 1 \] From this, we can express \( \sin \theta \) in terms of \( \cos \theta \) as follows:\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \] This is valid for acute angles where sine is positive.

Step 3: Substitute Sine in Cosecant Identity

Using the expression we found for \( \sin \theta \) in Step 2, substitute it into the cosecant identity: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\sqrt{1 - \cos^2 \theta}} \]

Key Concepts

Cosecant Identity

Understanding the cosecant identity is key to grasping trigonometric relationships. Cosecant, denoted as \( \csc \theta \), is a function that is closely related to the sine function. In simple terms, it is the reciprocal of sine. This implies that:

• If \( \sin \theta = \frac{3}{5} \), then \( \csc \theta = \frac{5}{3} \).
• The cosecant identity can be mathematically expressed as \( \csc \theta = \frac{1}{\sin \theta} \).

By using this identity, we can convert expressions involving sine into expressions involving cosecant and vice versa. It is particularly useful when simplifying trigonometric expressions into terms that may be more necessary for solving specific problems.
It's important to highlight that the cosecant function is undefined when \( \sin \theta = 0 \), since division by zero is undefined in mathematics. Hence, always be cautious of angles where this might happen.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental relationships in trigonometry. It connects the squares of the sine and cosine functions of an angle. The identity is expressed as:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)
• This identity holds true for any angle \( \theta \).

One practical application of this identity is its ability to express one trigonometric function in terms of another. For example, if you only know \( \cos \theta \), you can find \( \sin \theta \) by rearranging the identity formula:- \( \sin \theta = \sqrt{1 - \cos^2 \theta} \)When using this formula, it is crucial to consider the angle's quadrant, particularly for non-acute angles, as this affects the sign of the result. However, in acute angles, as discussed in the original solution, sine is always positive, simplifying the process considerably.

Reciprocal Functions

Trigonometry often involves dealing with reciprocal trigonometric functions, which are counterparts to the main trigonometric functions of sine, cosine, and tangent. These functions include cosecant, secant, and cotangent:

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

Reciprocal functions allow for more flexibility in manipulating expressions, especially when exact values of one function are known while the other needs to be determined. They provide alternative paths to simplify complex trigonometric equations. Additionally, understanding these functions is fundamental for solving more advanced trigonometric problems, as they often appear in various mathematical contexts. Mastering reciprocal functions ensures a deeper understanding of the interplay between different trigonometric identities and enhances problem-solving skills.