Problem 69 - Precalculus Essentials

Question

Use a right triangle to write each expression as an algebraic expression. Assume that \(x\) is positive and that the given inverse trigonometric function is defined for the expression in \(x\). $$\cot \left(\tan ^{-1} \frac{x}{\sqrt{3}}\right)$$

Step-by-Step Solution

Answer

The expression \(\cot (\tan ^{-1} \frac{x}{\sqrt{3}})\) simplifies to \(\frac{\sqrt{3}}{x}\)

Step 1: Understanding the trigonometric identity

Firstly, remember that the cotangent of an angle in a right triangle is the reciprocal of the tangent of the angle. So, it can also be expressed as \(\cot(\theta) = \frac{1}{\tan(\theta)}\). Now using this identity, \(\cot (\tan ^{-1} \frac{x}{\sqrt{3}})\) can be expressed as \(\frac{1}{\tan (\tan ^{-1} \frac{x}{\sqrt{3}})}\).

Step 2: Simplifying the function

The inverse tangent of \(x\) over \(\sqrt{3}\) represents an angle whose tangent is \(x/\sqrt{3}\). So, \(\tan (\tan ^{-1} \frac{x}{\sqrt{3}}) = \frac{x}{\sqrt{3}}\). Now, substitute this into the expression obtained in Step 1 to get \(\frac{1}{x/\sqrt{3}}\).

Step 3: Final Simplification

Lastly, compute the reciprocal of \(x/\sqrt{3}\) to simplify the expression. In division, the reciprocal of \(\frac{x}{\sqrt{3}}\) is \(\frac{\sqrt{3}}{x}\), which is the final simplified algebraic expression.

Key Concepts

Cotangent

The cotangent function, often shortened to 'cot', is one of the six fundamental trigonometric functions. In terms of a right-angled triangle, it is the ratio of the adjacent side to the opposite side. For any angle \( \theta \) in a right triangle, you can find the cotangent by using the formula \( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} \).
One vital relationship to remember is that cotangent is the reciprocal of the tangent function, which means \( \cot(\theta) = \frac{1}{\tan(\theta)} \). This identity simplifies the process of finding a cotangent, especially when dealing with inverse trigonometric functions like \( \tan^{-1} \) as in the given exercise.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides of the equation are defined. They play a crucial role in simplifying trigonometric expressions and solving trigonometric equations. Among the most frequently used identities are reciprocal identities like \( \tan(\theta) = \frac{1}{\cot(\theta)} \) and \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
In the given exercise, understanding reciprocal identities helps transform the cotangent of an inverse tangent into an algebraic expression. Knowing these identities allows students to approach trigonometry problems with confidence and agility.

Right Triangle Trigonometry

Right triangle trigonometry focuses on the relationships between angle measures and side lengths in right-angled triangles. The basic trigonometric ratios—sine, cosine, and tangent—along with their reciprocals (cosecant, secant, and cotangent), describe these relationships. Using the inverse trigonometric functions, such as \( \tan^{-1} \) (arctangent), allows us to find an angle when the opposite and adjacent sides are known.
This type of problem-solving is exemplified in the exercise, where interpreting \( \tan^{-1} \frac{x}{\sqrt{3}} \) involves recognizing that it represents an angle in a right triangle whose tangent is \( \frac{x}{\sqrt{3}} \). Facilitating a deeper connection with these concepts can significantly improve a student's ability to navigate more complex trigonometric problems.

Algebraic Expression Simplification

Algebraic expression simplification involves reducing expressions to their simplest form while maintaining their original value. This often implies getting rid of parentheses, combining like terms, and reducing fractions to lowest terms. It is vital in making expressions easier to understand or further manipulate within an equation.
In our exercise, the simplification comes into play when we take the reciprocal of \( \tan(\tan^{-1} \frac{x}{\sqrt{3}}) \) which gives us \( \frac{1}{\frac{x}{\sqrt{3}}} \) and ultimately simplifies to \( \frac{\sqrt{3}}{x} \). Such simplifications are key to making complex trigonometric expressions more manageable and highlight the interconnectedness between algebra and trigonometry.