Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\tan \theta, \text { given that } \cot \theta=-0.01$$

Step-by-Step Solution

Answer

\( \tan \theta = -100 \)

Step 1: Understand Cotangent and Tangent Relationship

The given value is \( \cot \theta = -0.01 \). We need to find \( \tan \theta \). Recall the reciprocal identity: \( \tan \theta = \frac{1}{\cot \theta} \).

Step 2: Substitute the Given Value

Substitute \( \cot \theta = -0.01 \) into the reciprocal identity: \[ \tan \theta = \frac{1}{-0.01} \].

Step 3: Calculate the Tangent Value

Evaluate the expression from Step 2: \[ \tan \theta = \frac{1}{-0.01} = -100 \]. Thus, \( \tan \theta = -100 \).

Key Concepts

Reciprocal Identities

Reciprocal identities are fundamental components of trigonometry. They establish relationships between trigonometric functions that allow us to interchangeably use different functions based on given conditions. The reciprocal identities focus on the idea that each of the six primary trigonometric functions has a counterpart that is its reciprocal.

To understand this better, let's explore some reciprocal identities:

• The reciprocal of sine (\( \sin \theta \)) is cosecant (\( \csc \theta \)). Therefore, \( \csc \theta = \frac{1}{\sin \theta} \).
• The reciprocal of cosine (\( \cos \theta \)) is secant (\( \sec \theta \)). So, \( \sec \theta = \frac{1}{\cos \theta} \).
• The reciprocal of tangent (\( \tan \theta \)) is cotangent (\( \cot \theta \)), resulting in \( \cot \theta = \frac{1}{\tan \theta} \). Conversely, \( \tan \theta = \frac{1}{\cot \theta} \).

These identities are especially useful for solving trigonometric equations, simplifying expressions, or converting between trigonometric forms. In the problem provided, knowing that \( \tan \theta = \frac{1}{\cot \theta} \) helped us find the tangent of an angle given its cotangent.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is one of the basic trigonometric functions. It relates to the angles and sides of right-angled triangles, and it can be expressed as a ratio of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

This function represents the ratio of the opposite side to the adjacent side in a right triangle. In the context of the unit circle, where the radius is always 1, \( \tan \theta \) is defined as the y-coordinate divided by the x-coordinate of a point on the circle.

• This ratio can result in positive or negative values depending on the quadrant in which the angle \( \theta \) lies.
• Tangents of angles become undefined when the cosine, or denominator, is zero, corresponding to vertical asymptotes in the tangent function curve.

In our original exercise, we used the property that \( \tan \theta = \frac{1}{\cot \theta} \). Given \( \cot \theta = -0.01 \), the tangent is easily calculated as \( -100 \). This value confirms \( \tan \theta \)'s responsiveness to both its reciprocal identity and its sign (positive or negative), depending on the initial condition.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is closely related to the tangent function. It is simply the reciprocal or inverse of the tangent: \( \cot \theta = \frac{1}{\tan \theta} \). In a right triangle, cotangent is defined as the ratio of the adjacent side to the opposite side.

In terms of sine and cosine, the cotangent can be expressed as\( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This indicates that the cotangent also embodies the interconnections between sine and cosine, just like the tangent does, but inverted:

• The cotangent changes sign depending on the quadrant containing the angle \( \theta \).
• Since \( \cot \theta \) is the reciprocal of \( \tan \theta \), it becomes undefined when sine, or its denominator, is zero.

In the given exercise, knowing \( \cot \theta = -0.01 \) helped us utilize and highlight the versatility of reciprocal identities. By simple inversion, we quickly calculated \( \tan \theta \), and reciprocally, used both functions to understand the reciprocal relationship in action.