Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$\tan 25^{\circ} 43^{\prime}$$

Step-by-Step Solution

Answer

\( \tan 25^\circ 43' = \cot(64^\circ 17') \).

Step 1: Introduction to Co-Functions

In trigonometry, co-functions are pairs of trigonometric functions that complement each other. The tangent function and the cotangent function are co-functions related through complementary angles, meaning the following identity holds: \( \tan(\theta) = \cot(90^\circ - \theta) \). We will use this relation to express \( \tan 25^\circ 43' \) in terms of its co-function.

Step 2: Convert Degree-Minute Format

First, we understand that \( 25^\circ 43' \) is given in a degree-minute format. To apply the co-function identity easily, we keep it as is but remember the angle \( \theta = 25^\circ 43' \) as we proceed with calculations.

Step 3: Find the Complementary Angle

To express \( \tan 25^\circ 43' \) in terms of \( \cot \), we need its complementary angle. The complementary angle is \( 90^\circ - 25^\circ 43' \). Calculating this gives us \( 90^\circ - (25^\circ 43') = 64^\circ 17' \).

Step 4: Use the Co-Function Identity

Apply the co-function identity \( \tan \theta = \cot(90^\circ - \theta) \). Substituting the values, we get \( \tan 25^\circ 43' = \cot(64^\circ 17') \).

Step 5: Conclusion

Thus, \( \tan 25^\circ 43' \) can be expressed in terms of its co-function as \( \cot(64^\circ 17') \).

Key Concepts

Co-Functions

The concept of co-functions in trigonometry revolves around pairs of functions that are linked through complementary angles. Co-functions share a special relationship where the value of one function at a certain angle is equal to the value of its co-function at the complementary angle. These pairs include:

• Sine (\(\sin\theta\)) and Cosine (\(\cos(90^\circ - \theta)\))
• Tangent (\(\tan\theta\)) and Cotangent (\(\cot(90^\circ - \theta)\))
• Secant (\(\sec\theta\)) and Cosecant (\(\csc(90^\circ - \theta)\))

Understanding co-functions helps in simplifying trigonometric expressions and solving equations efficiently. It allows mathematicians and students to express a trigonometric function in terms of its co-function using simple algebra.

Complementary Angles

Complementary angles are two angles that together add up to 90 degrees. This concept is crucial in the realm of co-functions as each function connects via their complementary angle. For example, if you have an angle \(\theta\), then its complement is \(90^\circ - \theta\). When working with trigonometric functions, we often use complementary angles to find co-function relationships.

In practical terms:

• If angle \(A\) is \(30^\circ\), its complementary angle \(B\) would be \(60^\circ\) because \(30^\circ + 60^\circ = 90^\circ\).
• Finding the complementary angle allows using the co-function identities, such as \(\tan(\theta) = \cot(90^\circ - \theta)\).

Applying this knowledge helps solve trigonometric problems by transforming angles into their complementary forms.

Cotangent

Cotangent is one of the fundamental trigonometric functions and is abbreviated as \(\cot\). It is the reciprocal of the tangent function and is defined as \(\cot\theta = \frac{1}{\tan\theta}\).

In terms of a right triangle, the cotangent of an angle is the ratio of the length of the adjacent side to the opposite side. Mathematically, this is expressed as:\[\cot\theta = \frac{\text{Adjacent}}{\text{Opposite}}\]
Cotangent is closely related to tangent through the co-function identity, where\(\cot(\theta) = \tan(90^\circ - \theta)\). This demonstrates their link through complementary angles. It underscores the concept that these functions can often be interchanged when solving trigonometric problems.

Tangent

The tangent function is one of the primary functions in trigonometry, denoted as \(\tan\). It represents the ratio of the opposite side to the adjacent side in a right-angled triangle, defined as:\[\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\]
Tangent is a key part of co-function identities. Since \(\tan\theta\) and \(\cot(90^\circ - \theta)\) are co-functions, they reflect the complementary angle relationship. This makes tangent flexible for transformation into cotangent expressions when angles are complementary.

Working with the tangent often involves calculating its cotangent co-function to simplify expressions or solve trigonometric questions effectively. Understanding these relationships provides a deeper insight into trigonometric principles and aids in mastering these mathematical concepts.