Problem 71 - Algebra and Trigonometry

Question

Express the exact value of each function as a single fraction. Do not use a calculator. $$ \text { If } \theta \text { is an acute angle and } \cot \theta=\frac{1}{4}, \text { find } \tan \left(\frac{\pi}{2}-\theta\right) $$

Step-by-Step Solution

Answer

The exact value of the function is \( \frac{1}{4} \).

Step 1: Recognize the Relationship

In trigonometry, the cotangent of an angle is the reciprocal of the tangent of the angle. Given that \( \cot \theta=\frac{1}{4} \), implies \( \tan \theta = 4 \).

Step 2: Use Trigonometric Identity

For the tangent of the complement of an angle, \( \tan \left(\frac{\pi}{2}-\theta\right) \), there is a trigonometric identity that states \( \tan \left(\frac{\pi}{2}-\theta\right) = \cot \theta \). This will be the relationship used to solve the problem. Substitute \( \cot \theta \) with \( \frac{1}{4} \).

Step 3: Carry out the Substitution

Substituting gives \( \tan \left(\frac{\pi}{2}-\theta\right)= \frac{1}{4} \). That's the solution!

Key Concepts

Cotangent

The cotangent of an angle, often abbreviated as 'cot', is one of the six primary trigonometric functions. Like its more commonly referenced counterpart, the tangent, the cotangent is an essential tool for solving various geometric and trigonometric problems. Cotangent specifically refers to the ratio of the adjacent side to the opposite side in a right-angled triangle. In mathematical terms, for an acute angle \( \theta \), it is defined as:
\[ \text{cot}(\theta) = \frac{\text{adjacent side}}{\text{opposite side}} = \frac{1}{\tan(\theta)} \]
where \( \tan(\theta) \) is the tangent of that angle, making cotangent its reciprocal. When using cotangent, it's vital to consider its domain and range, especially since it becomes undefined for angles where the opposite side length is zero (as division by zero is not permissible). Understanding the cotangent function is foundational to dealing with exercises involving the evaluation of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved variables where both sides of the equation are defined. These identities are invaluable for simplifying and solving trigonometry problems, as they help in transforming one form of expression into another, often making complicated calculations more manageable.

One of the most crucial identities in trigonometry involves the relationship between complementary angles:
\[ \tan \bigg(\frac{\text{π}}{2} - \theta\bigg) = \text{cot}(\theta) \]
This identity is a cornerstone in understanding the problem at hand, expressing the tangent of an angle's complement in terms of the cotangent of the original angle. Trigonometric identities can be further classified into categories such as Pythagorean identities, sum and difference identities, and double-angle identities, each serving different functions in simplifying and transforming trigonometric expressions.

Acute Angle

An acute angle is an angle that measures less than 90 degrees. In the context of trigonometry, trigonometric functions of acute angles are often encountered in right triangles, where one angle is 90 degrees, and the other two are acute. The properties of acute angles allow us to use trigonometric functions in a wide range of practical applications, including calculations involving length, height, and distance.

Characteristics of an acute angle make it fundamental in defining the basic trigonometric functions, as these functions evolved from the need to relate the sides of a right triangle to its non-right angles, which are necessarily acute. The fact that the exercise specifies the angle \( \theta \) as acute is significant, as trigonometric function values can differ vastly for non-acute angles, due to the periodic and cyclical nature of these functions.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are essential in various branches of mathematics, physics, engineering, and many other fields. The primary trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

Each of these functions takes an angle as an input and produces a single output, which is a ratio of two sides of a triangle, or, more generally, the coordinates of points on the unit circle. For right triangles, these functions are as follows:

• Sine represents the ratio of the opposite side to the hypotenuse.
• Cosine represents the ratio of the adjacent side to the hypotenuse.
• Tangent represents the ratio of the opposite side to the adjacent side.
• Cotangent, as previously mentioned, is the reciprocal of tangent.
• Secant is the reciprocal of cosine.
• Cosecant is the reciprocal of sine.

Understanding these functions and their interrelations is the key to solving a wide array of problems within trigonometry and beyond.