Question

Cofunctions. Express as a function of the complementary angle. $$\cot 63.2^{\circ}$$

Short answer

\(\cot(63.2^\circ) = \tan(26.8^\circ)\)

Step-by-step solution

Understand the Cofunction Identity for Cotangent

The cofunction identity states that the cotangent of an angle is the tangent of its complementary angle. In mathematical terms, \(\cot(\theta) = \tan(90^\circ - \theta)\). Since two angles are complementary if their sum is 90 degrees, we'll use this to express \(\cot 63.2^\circ\) as the tangent of its complementary angle.

Find the Complementary Angle

To find the complementary angle of \(63.2^\circ\), subtract it from \(90^\circ\): \(90^\circ - 63.2^\circ = 26.8^\circ\). This result is the complementary angle to \(63.2^\circ\).

Apply the Cofunction Identity

Use the cofunction identity from Step 1 on the complementary angle we found in Step 2: \(\cot(63.2^\circ) = \tan(90^\circ - 63.2^\circ) = \tan(26.8^\circ)\). Thus, \(\cot(63.2^\circ)\) can be expressed as \(\tan(26.8^\circ)\).

Key concepts

Cotangent

In trigonometry, the cotangent is one of the six fundamental trigonometric functions. It is the ratio of the adjacent side to the opposite side in a right-angled triangle. The formula for cotangent is \( \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} \), or equivalently, \( \text{cot}(\theta) = \frac{\text{cos}(\theta)}{\text{sin}(\theta)} \). This identity emphasizes that cotangent is inherently related to the tangent; it is simply its reciprocal.

For the angle \( 63.2^\text{o} \), the cotangent function would be providing information about a specific ratio in a triangle where \( 63.2^\text{o} \) is one of the non-right angles. Understanding cotangent is essential for solving various problems in geometry and trigonometry, especially when the problem involves complementary angles, as seen in the cofunction identities.

Complementary Angles

The idea of complementary angles is straightforward: if two angles add up to \( 90^\text{o} \), they are considered complementary. In a right-angled triangle, any of the non-right angles are complementary to each other.

This concept is not only crucial in geometry but also in trigonometry, where it plays a major role in defining the relationship between trigonometric functions of angles and their complements. Relatedly, in the context of the exercise, when we say \( \text{cot}(\theta) = \text{tan}(90^\text{o} - \theta) \), we are using the cofunction identity that links cotangent and tangent through their complementary angles.

Trigonometric Identities

Trigonometry is rich with trigonometric identities, which are equations that hold true for all possible values of the involved variables. These identities are crucial for simplifying expressions and solving equations involving trigonometric functions.

The cofunction identities, which include the one used in the exercise, are a group of trigonometric identities. These identities express the sine, cosine, tangent, cotangent, secant, and cosecant of an angle in terms of the complementary angle. Understanding these identities allows you to connect various trigonometric functions and simplify problems to more manageable terms.

Tangent

The tangent of an angle in a right triangle is the ratio of the side opposite to the angle to the side adjacent to the angle. In formulaic terms, this is \( \text{tan}(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)} \).

It's a function often used to determine the slope of an angle or to find the height of an object when the angle of elevation and distance to the object are known. The tangent function has a direct relationship with the cotangent function, being their reciprocals, which is a key element in understanding and applying cofunction identities as shown in the example where \( \text{cot}(63.2^\text{o}) = \text{tan}(26.8^\text{o}) \) following from \( \text{tan}(90^\text{o} - \theta) \).