Problem 8 - Algebra 2

Question

Evaluate each expression. Write your answer in exact form. Suppose tan \(\theta=-\frac{4}{3},\) Find \(\cot \theta\)

Step-by-Step Solution

Answer

The value of \( \cot \theta \) is \( -\frac{3}{4} \)

Step 1: Understand the relationship

First, recognize the relationship between tangent and cotangent. We know that \(\cot \theta = \frac{1}{\tan \theta}\). So, to find \(\cot \theta\), we need to take the reciprocal of the given \(\tan \theta\).

Step 2: Plug in the given value

We are given that \(\tan \theta = -\frac{4}{3}\). So, we substitute this value into our equation from Step 1. Then we get \(\cot \theta = \frac{1}{-\frac{4}{3}}\).

Step 3: Simplify the expression

Perform division by using the 'change-flip' rule (change the division to a multiplication by flipping the fraction following the division sign). Then, \(\cot \theta = \frac{1}{1} * \frac{3}{-4} = -\frac{3}{4}\).

Key Concepts

Tangent

Tangent is one of the most fundamental trigonometric functions. This function is related to the tangent line of a circle, but in trigonometry, it connects an angle in a right triangle to the ratio of specific side lengths. Specifically, the tangent of an angle, often denoted as \( \tan \theta \), equals the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. For example, if \( \theta \) is an angle in a right triangle, with the opposite side having a length of 4 units and the adjacent side having a length of 3 units, the tangent of this angle would be:

• Formula: \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)
• Example: \( \tan \theta = \frac{4}{3} \)

The tangent can be positive or negative, depending on the quadrant in which the angle is situated. In the given exercise, \( \tan \theta \) is \(-\frac{4}{3}\), indicating that the angle is in either the second or fourth quadrant, where the tangent function takes a negative value.

Cotangent

Cotangent is another trigonometric function, closely related to tangent. It is essentially the reciprocal of the tangent function. Instead of involving the ratio of opposite to adjacent side, like tangent, cotangent uses the ratio of the adjacent side to the opposite side. This relation is particularly useful, as it allows for straightforward calculations involving angles when the tangent is known. It can be expressed as:

• Formula: \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \)
• Reciprocal: \( \cot \theta = \frac{1}{\tan \theta} \)

To find the cotangent when the tangent is provided, simply take the reciprocal of the given tangent value. For instance, in our case: if \( \tan \theta = -\frac{4}{3} \), then \( \cot \theta = \frac{1}{-\frac{4}{3}} \). By performing this calculation, we find \( \cot \theta = -\frac{3}{4} \).

Reciprocal Function

Reciprocal functions are an important concept in mathematics, particularly in trigonometry, where they form the basis for working with different trigonometric identities. The idea of a reciprocal is simple: it is the inverse of a number or expression, which, when multiplied by the original number, equals one. In simpler terms, if \( a \times b = 1 \), then \( b \) is the reciprocal of \( a \).

• Definition: \( a^{-1} = \frac{1}{a} \)
• Example: The reciprocal of \( \frac{4}{3} \) is \( \frac{3}{4} \)

In trigonometry, this principle is applied to functions. For the tangent function \( (\tan \theta) \), the reciprocal is the cotangent \( (\cot \theta) \). By understanding this relationship, students can easily interchange between these two functions and simplify complex expressions. Recognizing these reciprocal relationships helps unlock deeper comprehension of trigonometric identities and problem-solving.