Problem 40 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ \cot \alpha \sin \alpha=\cos \alpha $$

Step-by-Step Solution

Answer

The identity \(\cot \alpha \sin \alpha = \cos \alpha\) holds true for \(0<\theta<\pi / 2)\). It's verified by transforming \(\cot \alpha\) into \( \frac{1}{\tan \alpha}\) and further simplification.

Step 1: Identify Trigonometric Identity

\(\cot \alpha\) can be expressed as the reciprocal of \(\tan \alpha\), thus we can rewrite the left side of the equation as \( \frac{\sin \alpha}{\tan \alpha}\).

Step 2: Express Tangent as Sine over Cosine

Knowing that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \), we can substitute this into the equation, giving us \( \frac{\sin \alpha}{\frac{\sin \alpha}{\cos \alpha}}\).

Step 3: Simplify the Equation

The \(\sin \alpha\) in the numerator and denominator cancel out to leave \(\cos \alpha\) on the left side of the equation, which agrees with the right side of the equation. Therefore, we have the equation \( \cos \alpha = \cos \alpha \).

Key Concepts

Cotangent

Cotangent is one of the lesser-known trigonometric functions but plays a crucial role as the reciprocal of tangent. In terms of sine and cosine, cotangent (\( ext{cot} \alpha\)) is defined as:

• \[\cot \alpha = \frac{\cos \alpha}{\sin \alpha}.\]

This identity is extremely useful when simplifying trigonometric expressions, as it allows us to interchange between the different functions. When solving problems or converting expressions, recognizing this relationship can make complex equations more manageable. As we explore other trigonometric functions, you'll find that they often intertwine, allowing for various approaches in problem-solving.

Tangent

The tangent function is a fundamental aspect of trigonometry and is often encountered in a variety of mathematical problems. It is defined as the ratio of the sine to the cosine of a given angle:

• \[\tan \alpha = \frac{\sin \alpha}{\cos \alpha}.\]

Knowing this identity is key to expressing tangent in terms of sine and cosine, making it easier to integrate it into equations. In our specific problem, we transformed the equation by substituting the tangent identity, paving the way to simplification. Grasping this conversion helps in understanding how tangent interacts within various trigonometric contexts, enabling deeper insights into relationships between angles and sides in triangles.

Sine

The sine function is one of the primary trigonometric functions frequently used to solve problems involving right triangles and circles. Defined as the ratio of the opposite side to the hypotenuse in a right triangle, sine connects the angle with the triangle's sides:

• \[\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}.\]

In the equation from our problem, sine was a key component that interacted with cotangent and tangent, providing a pathway to simplify and prove the given trigonometric identity. Understanding sine's role not only helps in solving equations but also enhances comprehension of wave functions and their applications in real-world scenarios.

Cosine

Cosine is commonly paired with sine and comes as another trigonometric staple in analyzing angles. It is defined in a right triangle as the ratio of the adjacent side to the hypotenuse:

• \[\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}.\]

The cosine function was the result of simplifying the original trigonometric identity, illustrating its integral connection with other functions. Understanding cosine's properties allow for a straightforward simplification of identities and equations involving trigonometric expressions. In many practical applications, cosine helps to model oscillatory movements and patterns, cementing its importance both in theoretical and applied mathematics.