Problem 10 - Algebra 2

Question

Find the value of each expression. \(\tan \theta,\) if \(\cot \theta=2 ; 0^{\circ} \leq \theta<90^{\circ}\)

Step-by-Step Solution

Answer

\(\tan \theta = \frac{1}{2}\).

Step 1: Understand Reciprocal Identities

Recall that the cotangent and tangent functions are reciprocals of each other. Therefore, we can express the tangent of an angle \(\theta\) in terms of the cotangent as \(\tan \theta = \frac{1}{\cot \theta}\).

Step 2: Substitute the Given Value

Since we know that \(\cot \theta = 2\), we can substitute this value into the reciprocal identity for tangent to get \(\tan \theta = \frac{1}{2}\).

Step 3: Confirm the Range of \(\theta\)

Check the range given for \(\theta\), which is \(0^{\circ} \leq \theta < 90^{\circ}\). This range corresponds to the first quadrant where both \(\tan \theta\) and \(\cot \theta\) are positive, confirming that \(\tan \theta = \frac{1}{2}\) is valid.

Key Concepts

Reciprocal Identities

In trigonometry, reciprocal identities offer a handy way to express one trigonometric function in terms of another. Understanding these relationships can simplify problem-solving when dealing with triangles or periodic functions. For the tangent and cotangent functions, their reciprocal relationship is expressed as:

• \( \tan \theta = \frac{1}{\cot \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities allow us to interchangeably use one function when the other is given. This makes solving expressions much easier, especially when a direct computation might be complex or impossible. In our original problem, since the angle \( \theta \) is in the range \( 0^{\circ} \leq \theta < 90^{\circ} \), both functions are straightforwardly positive. This knowledge is crucial as it confirms the result of using reciprocal identities correctly, ensuring we apply the correct sign.

Cotangent Function

The cotangent function, represented as \( \cot \theta \), is one of the principal trigonometric functions. It is particularly useful in problems involving right-angled triangles and periodic phenomena. The cotangent of an angle is defined as the reciprocal of its tangent, which can be mathematically expressed as follows: \[ \cot \theta = \frac{1}{\tan \theta} \] Alternatively, in terms of sine and cosine: \[ \cot \theta = \frac{\cos \theta}{\sin \theta} \] The function is undefined when \( \theta = 0^{\circ}, 180^{\circ}, 360^{\circ}, ... \), since the tangent of these angles is zero, leading to division by zero. In this particular exercise, we deal with \( \cot \theta = 2 \), which means that the tangent \( \tan \theta = \frac{1}{2} \). Thus, \( \cot \theta \) provides a direct pathway to finding \( \tan \theta \), grounding our understanding of reciprocal identities in practice.

Tangent Function

In trigonometry, the tangent function, denoted as \( \tan \theta \), plays a fundamental role. It is defined as the ratio of the sine to the cosine of an angle: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \] The tangent function is periodic with a period of \(180^{\circ} \) or \( \pi \) radians, meaning it repeats its values every \(180^{\circ} \). For angles in the range \(0^{\circ} \leq \theta < 90^{\circ}\), \( \tan \theta \) is always positive. This is important in confirming that, when using reciprocal identities, our calculations remain consistent with the expected positive value in this quadrant. In the provided problem, where \( \cot \theta = 2 \), using the reciprocal identity \( \tan \theta = \frac{1}{\cot \theta} \) simplifies the computation to \( \tan \theta = \frac{1}{2} \). This example demonstrates the practicality of knowing both the definition and properties of the tangent function.