Problem 28 - Calculus of a Single Variable

Question

Write the expression in algebraic form. (Hint: Sketch a right triangle, as demonstrated in Example 3.) \(\cos (\operatorname{arccot} x)\)

Step-by-Step Solution

Answer

The algebraic form of the given trigonometric expression is \(\cos (\operatorname{arccot} x) = \frac{x}{\sqrt{x^2 + 1}}\).

Step 1: Setup Right Triangle

Consider a right triangle. We let the angle \(A\) be such that \(\cot(A) = x\). Since cotangent is the ratio of the adjacent side over the opposite side in a right triangle, we can assign the lengths of the sides of the triangle accordingly. Let the side adjacent to angle \(A\) be \(x\) (because this serves as our 'base' in the cotangent ratio), the side opposite to angle \(A\) be \(1\) (this will serve as our 'height' in the ratio. We then calculate the hypotenuse using the Pythagorean theorem.

Step 2: Calculate Hypotenuse

According to the Pythagorean theorem, in a right triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, the length of the hypotenuse \(h\) can be found using the formula \(h = \sqrt{base^2 + height^2} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1}\).

Step 3: Apply Cosine Function

We then apply the cosine function to angle \(A\). In a right triangle, the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle and the length of the hypotenuse. Therefore, \(\cos(A) = \frac{x}{\sqrt{x^2 + 1}}\).

Key Concepts

Arccotangent

Arccotangent, often denoted as \( \operatorname{arccot} \), is the inverse function of the cotangent. When you encounter \( \operatorname{arccot} \), it refers to finding an angle whose cotangent value is a specific number. This is useful in various trigonometric problems where you want to determine the angle based on the cotangent ratio.
For example, if you have \( \operatorname{arccot}(x) \), it means you are looking for an angle \( A \) such that \( \cot(A) = x \). This is the starting point for many trigonometry problems, including those that require you to use other trigonometric functions like cosine on that angle.

Right Triangle

A right triangle is a triangle with one angle measuring 90 degrees. Each side of the triangle has a specific name depending on its position relative to the angle of interest. Knowing how these sides relate helps us solve many trigonometry problems.

• Adjacent side: The side next to the angle you are considering.
• Opposite side: The side opposite to the angle.
• Hypotenuse: The longest side of the triangle, opposite the right angle.

Understanding these terms helps in setting up trigonometric ratios, which are essential for solving expressions involving triangles.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, used to find the length of a side in a right triangle. It states that for a right triangle, the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides \( a \) and \( b \).
\[ c^2 = a^2 + b^2\]
For the problem at hand, if you know the lengths of the two sides are \( x \) and \( 1 \), you can easily determine the hypotenuse as \( \sqrt{x^2 + 1} \). This calculation extends the ability to use the trigonometric functions on angles within the triangle.

Cosine Function

The cosine function relates the length of the adjacent side to the hypotenuse in a right triangle. In terms of the angle \( A \), cosine is defined as:\[ \cos(A) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}\]
In our case, for an angle \( A \) where \( \cot(A) = x \), and we've found the hypotenuse to be \( \sqrt{x^2 + 1} \), the cosine of \( A \) becomes:\[ \cos(A) = \frac{x}{\sqrt{x^2 + 1}}\]
This allows us to convert the trigonometric expression into a more usable algebraic form, which simplifies many calculations in geometry and trigonometry.