Problem 70 - Algebra and Trigonometry

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 \(x\). $$ \cot \left(\tan ^{-1} \frac{x}{\sqrt{2}}\right) $$

Step-by-Step Solution

Answer

The value of the expression \(\cot(\tan^{-1}(x/\sqrt{2}))\) is \( \sqrt{2}/x\).

Step 1: Illustrate with a Right Triangle

The plugin to trigonometric equation is done by drawing a right triangle. Consider a right triangle, where the opposite side is \(x\) and the adjacent side is \(\sqrt{2}\). The hypotenuse can be found by applying the Pythagorean theorem: \(\sqrt{x^2+2}\).

Step 2: Invoke the Inverse Tangent Function

The inverse tangent function of \(x/\sqrt{2}\) can be represented as the angle of the triangle: \(\tan^{-1}(x/\sqrt{2}) = \theta\). Therefore, this configuration gives us \(\tan(\theta) = x/\sqrt{2}\).

Step 3: Find the Cotangent of Angle

Since the cotangent of an angle is the reciprocal of the tangent of the angle, we get: \(\cot(\theta) = 1/(\tan(\theta)) = \sqrt{2}/x\). Therefore, \(\cot(\tan^{-1}(x/\sqrt{2})) = \sqrt{2}/x\).

Key Concepts

Algebraic Expressions

An algebraic expression combines numbers, operations such as addition and multiplication, and variables like \( x \). In this exercise, we're working with an expression that involves a trigonometric function and its inverse. Transforming a trigonometric expression into an algebraic one involves simplifying using identities or the geometry of a right triangle. This can make solving problems that include angles and ratios much simpler. For example:

• Algebraic expressions help us convert a complex trigonometric function into something more recognizable, like \( \sqrt{2}/x \).
• This process allows us to find a simpler form using known relationships and functions.

By interpreting and manipulating the symbolic representations, we can unlock solutions to trigonometric problems using a straightforward, algebraic approach.

Right Triangle

A right triangle is a fundamental geometry shape in mathematics, specifically important in trigonometry. It has one angle of 90 degrees. In this particular problem, considering a right triangle makes it easier to relate the sides of the triangle to the trigonometric function given.

• Here, we consider a triangle where one leg is \( x \), another is \( \sqrt{2} \), and the hypotenuse is derived using the Pythagorean theorem.
• Using this triangle allows us to visually interpret and calculate trigonometric ratios like tangent and cotangent.

By drawing and understanding this triangle, we bridge the gap between algebraic and trigonometric expressions effortlessly.

Pythagorean Theorem

The Pythagorean Theorem is essential in deriving the hypotenuse of a right triangle. It states that in such triangles, the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Mathematically, it is expressed as:\[ a^2 + b^2 = c^2 \]In our exercise, we calculate the hypotenuse as:

• Given sides: \( x \) and \( \sqrt{2} \)
• Hypotenuse: \( \sqrt{x^2 + 2} \)

This calculation is vital because it provides the third side of the triangle, which helps in figuring out other trigonometric ratios. Knowing this relation provides a solid foundation for visualizing and solving geometric and trigonometric problems.

Trigonometric Ratios

Trigonometric ratios are the relationships between the angles and sides of a right triangle. They include sine, cosine, tangent, cotangent, secant, and cosecant. In this exercise, we focus on tangent and cotangent.

• For angle \( \theta \) in our triangle, \( \tan(\theta) = \frac{x}{\sqrt{2}} \).
• Once \( \tan(\theta) \) is known, \( \cot(\theta) \) becomes its reciprocal, \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\sqrt{2}}{x} \).

These ratios are powerful tools in trigonometry as they help derive unknown side lengths and angles in triangles, transforming complex trigonometric expressions into more manageable forms.