Problem 64 - Precalculus: A Concise Course

Question

Find the exact value of the expression. (Hint: Sketch a right triangle.) $$ \cot \left(\arctan \frac{5}{8}\right) $$

Step-by-Step Solution

Answer

The exact value of the expression \( \cot \left( \arctan \frac{5}{8} \right) \) is \( \frac{8}{5} \).

Step 1: Interpretation of the arctangent function

The function \(\arctan(x)\) gives the angle whose tangent is x. Here it is \(\arctan \frac{5}{8}\), so an angle \(\theta\) is needed in which \(\tan(\theta) = \frac{5}{8}\). At this point it is known that tangent is the ratio of the opposite to the adjacent in a right angle triangle.

Step 2: Construction of the right triangle

Construct a right triangle where the opposite side is 5 and the adjacent side is 8. The angle \(\theta = \arctan \frac{5}{8}\) is the angle that the adjacent side (8 units long) makes with the baseline.

Step 3: Calculation of the cotangent

Cotangent is the reciprocal of tangent. In terms of a right triangle, cotangent is the ratio of the adjacent to the opposite. In the right triangle, the adjacent side is of length 8 units and the opposite side is of length 5 units, so \(\cot (\theta)\) can be expressed as the ratio \( \frac{8}{5} \).

Key Concepts

Right Triangles

A right triangle is a type of triangle that has one angle exactly equal to 90 degrees. This characteristic makes it unique and particularly useful in trigonometry. Each right triangle is composed of three sides: the hypotenuse, which is the longest side and opposite the right angle; the opposite side, which is the side opposite the angle of interest; and the adjacent side, which is the side that forms the angle of interest with the hypotenuse.
Understanding these relationships in a right triangle helps solve various trigonometric problems. For instance, the tangent of an angle is the ratio of the opposite side to the adjacent side, while the cotangent is the reciprocal of the tangent, making it the ratio of the adjacent side to the opposite side.

Arctangent

The arctangent function, denoted as \( ext{arctan}\), is an inverse trigonometric function that is used to find the angle when the tangent value is known. Specifically, if you have a tangent value of a fraction, such as \(rac{5}{8}\), applying the arctangent function will yield an angle \(\theta\) such that \(\tan(\theta) = \frac{5}{8}\).
This concept is useful when determining angles in a right triangle where the sides are given, but the angle is not. In our example, knowing that \( heta = ext{arctan} \frac{5}{8} \) helps us understand that the opposite side is 5 and the adjacent side is 8, allowing us to construct a right triangle to solve further problems.

Cotangent

Cotangent, abbreviated as \( ext{cot}\), is one of the six trigonometric functions. It is the reciprocal of the tangent function. While tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle, cotangent is defined as the ratio of the adjacent side to the opposite side. Thus, \( ext{cot} \theta = \frac{1}{\tan \theta}\).
In practical terms, if you know the tangent of an angle, you can easily find the cotangent by taking the reciprocal of the tangent value. In our example, because \( \tan(\theta) = \frac{5}{8} \), the cotangent of \( \theta \) is \( \frac{8}{5} \). This concept simplifies the process of solving trigonometric problems involving angles without directly measuring them.