Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Complete the identity. $$\cot \left(90^{\circ}-\theta\right)=\square$$

Step-by-Step Solution

Answer

\(\cot(90^{\circ}-\theta) = \tan(\theta)\)

Step 1: Understand the Cotangent Function

The cotangent function is the reciprocal of the tangent function. It is often represented as \(\cot(\theta) = \frac{1}{\tan(\theta)}\). However, this does not immediately help in this situation.

Step 2: Apply the Co-Function Identity

The co-function identity of the cotangent function can be applied. For any acute angle \(\theta\), the co-function identity for the cotangent function is \(\cot(90^{\circ} - \theta) = \tan(\theta)\).

Step 3: Finalizing the Identity

We can affirm now that \(\cot(90^{\circ} - \theta)\) equals \(\tan(\theta)\). Hence, the square in the exercise should be replaced by \(\tan(\theta)\).

Key Concepts

Cotangent Function

The cotangent function, denoted as \textbf{cot}, is an important trigonometric function that is the reciprocal of the tangent function. It is defined as \( \cot(\theta) = \frac{1}{\tan(\theta)} \) or equivalently as \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \). This means for any angle \( \theta \), the cotangent of that angle is the ratio of the adjacent side to the opposite side of a right-angled triangle.

Understanding the cotangent function involves recognizing that it continues to exist and provide meaningful values even beyond acute angles (less than \(90^\circ\)). For example, it produces different signs depending on the quadrant in which the angle lies. It is periodic with a period of \(180^\circ\) or \(\pi\) radians, reflecting inherent symmetries in the unit circle. This property can apply when solving problems, especially when angles are beyond the first quadrant.

Moreover, the cotangent function has its own graph, which is different from that of the tangent function—a series of decreasing curves that pass through each point where \( \theta \) is an integer multiple of \(180^\circ\). In practical applications, knowing the behavior of the cotangent function can be vital in fields ranging from engineering to physics.

Tangent Function

The tangent function, written as \textbf{tan}, is another fundamental trigonometric function that represents the ratio of the opposite side to the adjacent side in a right-angled triangle with respect to a given angle \( \theta \). It can be defined as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) or the reciprocal of cotangent \( \cot(\theta) \).

The value of the tangent function varies significantly as the angle \( \theta \) approaches \(90^\circ\) (\(\pi/2\) radians) or \(270^\circ\) (\(3\pi/2\) radians), where it becomes undefined due to the cosine of the angle becoming zero. In the context of the unit circle, \( \tan(\theta) \) is the slope of the line segment from the origin to a point on the unit circle.

Furthermore, the graph of the tangent function shows its periodicity of \(180^\circ\) (\(\pi\) radians) and its characteristic asymptotes where the function is undefined. This information is used to solve a wide variety of trigonometric equations and model periodic phenomena such as sound waves and light waves.

Co-Function Identities

Co-function identities are a set of trigonometric identities that reflect the symmetric properties of the trigonometric functions. In essence, they state that certain pairs of trigonometric functions (sine and cosine, tangent and cotangent, secant and cosecant) of complementary angles (angles that add up to \(90^\circ\) or \(\pi/2\) radians) are equal.

To elaborate, the co-function identities for tangent and cotangent are:

• \[\cot(90^\circ - \theta) = \tan(\theta)\]
• \[\tan(90^\circ - \theta) = \cot(\theta)\]

These identities show that for any angle \( \theta \), the cotangent of its complement (\[90^\circ - \theta\]) is equal to the tangent of \( \theta \), and vice versa. This fact is widely useful in simplifying expressions and solving trigonometric problems that involve complementary angles.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all allowable values of the involved variables. They are invaluable tools in solving trigonometric equations and proving relationships between angles and lengths in geometric figures.

Some of the most frequently used trigonometric identities include Pythagorean identities, quotient identities, and reciprocal identities, in addition to the co-function identities discussed previously. They are the backbone of trigonometry and are used to simplify complex trigonometric expressions, find angle measures, compute lengths, and in many cases, are applied in solving real-world problems.

An example of a basic trigonometric identity is the Pythagorean identity, which relates the square of the sine and cosine functions: \[\sin^2(\theta) + \cos^2(\theta) = 1\]. Knowing and understanding these identities is essential for anyone studying trigonometry, as they provide a way to maneuver between different trigonometric functions and solve problems more efficiently.