Problem 43 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

\( \cot \theta = \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} \)

Step 1: Recalling Trigonometric Identities

First, let's recall that the cotangent function is the reciprocal of the tangent function. Therefore, the basic identity is given by \( \cot \theta = \frac{1}{\tan \theta} \).

Step 2: Express Tangent in Terms of Sine and Cosine

The tangent function is related to sine and cosine by the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Using this identity, we can rewrite \( \cot \theta \) in terms of sine and cosine.

Step 3: Substitute in Cotangent Identity

Substituting the expression of tangent in terms of sine and cosine, we get \( \cot \theta = \frac{1}{\frac{\sin \theta}{\cos \theta}} \).

Step 4: Simplify the Expression

To simplify \( \frac{1}{\frac{\sin \theta}{\cos \theta}} \), multiply the numerator and the denominator by \( \cos \theta \). Thus, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 5: Express Cosine in Terms of Sine

Use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to express \( \cos \theta \) in terms of \( \sin \theta \), which gives \( \cos \theta = \sqrt{1 - \sin^2 \theta} \).

Step 6: Final Expression in Terms of Sine

Substitute \( \cos \theta = \sqrt{1 - \sin^2 \theta} \) into \( \frac{\cos \theta}{\sin \theta} \) to express \( \cot \theta \) solely in terms of \( \sin \theta \). Therefore, we have \( \cot \theta = \frac{\sqrt{1 - \sin^2 \theta}}{\sin \theta} \).

Key Concepts

Cotangent Function

The cotangent function, usually denoted as \( \cot \theta \), is one of the six primary trigonometric functions. It is defined as the reciprocal of the tangent function. So, if you know the tangent of an angle \( \theta \), you can easily find \( \cot \theta \) as follows:

• \( \cot \theta = \frac{1}{\tan \theta} \)

Since tangent is the ratio of the sine to the cosine of an angle, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), the cotangent can be rewritten using this relationship. By substituting for tangent, the identity becomes:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This expression highlights how cotangent directly ties back to the primary sine and cosine functions, providing a straightforward way to compute it from these more familiar trigonometric ratios.

Sine and Cosine Relationship

The sine and cosine functions are closely intertwined. They're the fundamental building blocks of trigonometry. Every trigonometric expression can often be rewritten in terms of only sine and cosine.

• \( \sin \theta \) represents the "opposite" side in a right-angled triangle relative to an angle \( \theta \).
• \( \cos \theta \) represents the "adjacent" side relative to the same angle.

The relationship between sine and cosine not only forms the basis for the definition of tangent and cotangent but also gives rise to many identities that simplify trigonometric expressions.For example, remember:

• The tangent function is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• Therefore, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Understanding these interconnections is vital to mastering trigonometry. It allows you to manipulate and simplify a wide range of trigonometric expressions effectively.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry, linking the sine and cosine functions in a fundamental way. It states:

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

This identity derives from the Pythagorean theorem, describing the relationship between the sides of a right triangle. In trigonometric terms, it ensures that for any angle, the square of the sine plus the square of the cosine equals one.This identity is useful for expressing one trigonometric function in terms of another. For instance, if you only have \( \sin \theta \) and need \( \cos \theta \), you can rearrange and solve:

• \( \cos^2 \theta = 1 - \sin^2 \theta \)
• Thus, \( \cos \theta = \sqrt{1 - \sin^2 \theta} \)

By understanding and applying the Pythagorean identity, you gain a powerful tool to navigate between the sine and cosine functions, aiding in the simplification of complex trigonometric expressions.