Problem 43 - Algebra 2

Question

Express the first trigonometric function in terms of the second. $$ \cot \theta, \csc \theta $$

Step-by-Step Solution

Answer

The expression of \(\cot \theta\) in terms of \(\csc \theta\) is \(\sqrt{\csc^2 \theta - 1}\).

Step 1: Express Cotangent in Terms of Sine and Cosine

We start by expressing the cotangent function, \(\cot \theta\), in terms of cosine and sine. \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Step 2: Replace Sine with its Reciprocal

Next, we replace \(\sin \theta\) with \(\frac{1}{\csc \theta}\). So, our \(\cot \theta\) becomes \(\cot \theta = \frac{\cos \theta}{\frac{1}{\csc \theta}} = \cos \theta * \csc \theta\).

Step 3: Replace Cosine with Cosecant

Remember that, \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). Since \(\sin \theta = \frac{1}{\csc \theta}\), we can write \(\cos \theta = \sqrt{1 - \left(\frac{1}{\csc \theta}\right)^2} = \sqrt{\frac{\csc^2 \theta - 1}{\csc^2 \theta}} = \frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta}\). Therefore, \(\cot \theta = \frac{\sqrt{\csc^2 \theta - 1}}{\csc \theta} * \csc \theta = \sqrt{\csc^2 \theta - 1}\).

Key Concepts

Cotangent Function

The cotangent function, often symbolized as \( \cot \theta \), is a trigonometric ratio that is the reciprocal of the tangent function. Essentially, it gives us the ratio of the adjacent side to the opposite side in a right-angled triangle. This can be interpreted as:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• In terms of sine and cosine, it is expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Breaking down this formula further, the cotangent function highlights the pivotal relationship between the cosine and sine functions. Rather than focusing on finding the angle’s tangent directly, it leverages the fundamental trigonometric properties of cosine and sine. This makes cotangent a unique tool, primarily valuable when these components \( (\cos \theta \text{ and } \sin \theta) \) are readily available for calculations.

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is another reciprocal trigonometric function. It is specifically the reciprocal of sine, providing a relation reflecting the length dynamics of the hypotenuse to the opposite side in a triangle:

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

Using the cosecant function effectively transforms direct calculations of \( \sin \theta \) into finding the reciprocal, thus simplifying various trigonometric expressions or equations. For example, in the given solution, \( \sin \theta \) is replaced with its reciprocal \( \frac{1}{\csc \theta} \), simplifying the process of finding expressions for other trigonometric functions such as cotangent.

Sin and Cos Relationships

Sine and cosine functions form the foundational core of trigonometry. Understanding the relationship between these functions is essential, especially when translating between different trigonometric equations. Here’s how it works:

• The Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Express Cosine in terms of Sine: \( \cos \theta = \sqrt{1 - \sin^2 \theta} \)

In the original step-by-step solution, this identity is crucial. When replacing \( \cos \theta \), \( \sin \theta \) is expressed using its reciprocal function (cosecant). The substitution results in \( \cos \theta = \sqrt{1 - \left(\frac{1}{\csc \theta}\right)^2} \), converting the cosine into a form using cosecant expressions. This ability to express one trigonometric function in terms of another is vital for complex problem-solving and advanced trigonometric manipulations.