Problem 54 - Algebra 2

Question

ACT/SAT If the cotangent of angle \(\theta\) is \(1,\) then the tangent of angle \(\theta\) is $$ \begin{array}{lll}{\mathbf{A}-1} & {} & {\mathbf{C}} & {1} \\ {\mathbf{B}} & {0} & {} & {\mathbf{D}} & {3}\end{array} $$

Step-by-Step Solution

Answer

C) 1

Step 1: Understanding Cotangent

The cotangent of an angle \( \theta \) is defined as the reciprocal of the tangent of that angle. Therefore, for \( \cot(\theta) = 1 \), we have the relationship \( \tan(\theta) = \frac{1}{\cot(\theta)} \).

Step 2: Finding the Tangent

Since \( \cot(\theta) = 1 \), it follows that \( \tan(\theta) = \frac{1}{1} = 1 \).

Step 3: Selecting the Correct Answer

With \( \tan(\theta) = 1 \), we compare the result to the given answer choices: A) \(-1\), B) \(0\), C) \(1\), D) \(3\). The correct answer is C) \(1\).

Key Concepts

Cotangent

The cotangent of an angle, often abbreviated as "cot," is a fundamental trigonometric concept. It relates the lengths of the adjacent side to the opposite side in a right-angled triangle. Specifically, cotangent is the reciprocal of the tangent function. This means if you know the tangent of an angle, you can easily find the cotangent by taking the reciprocal.When you hear that the cotangent of an angle \( \theta \) is equal to 1, it implies a special relationship. It means the lengths of the opposite and adjacent sides are equal. This only happens in specific situations in the unit circle, such as at angles 45° (or \( \pi/4 \) radians).Remember:

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• Cotangent is undefined for angles where sine is zero, such as 0° or 180°.

Understanding cotangent can be crucial when solving trigonometric equations, especially when converting between different trigonometric functions.

Tangent

Tangent, commonly denoted as "tan," is another essential trigonometric ratio. It compares the length of the opposite side to the length of the adjacent side in a right-angled triangle. The tangent function is crucial for describing angles and their relationships, especially in geometry and calculus.When \( \tan(\theta) = 1 \), it indicates that the opposite and adjacent sides of the triangle are of equal length. This is precisely the same condition for when \( \theta \) is 45° (or \( \pi/4 \) radians), a very common angle in trigonometry due to its symmetry.Key points to remember:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
• The value of the tangent function can be any real number, and it is undefined at angles where cosine equals zero, such as 90° and 270°.

Understanding tangent and its properties can greatly aid in solving math problems involving trigonometry and can also help in visualizing angles in terms of their practical applications.

Reciprocal

The concept of reciprocal is a common mathematical operation that is often used in trigonometry. It plays a pivotal role in understanding how different trigonometric functions relate to each other.A reciprocal of a number is simply 1 divided by that number. In trigonometry, almost every primary function (sine, cosine, and tangent) has a corresponding reciprocal function (cosecant, secant, and cotangent).For instance:

• The reciprocal of tangent is cotangent: \( \cot(\theta) = \frac{1}{\tan(\theta)} \).
• When given \( \cot(\theta) = 1 \), finding \( \tan(\theta) \) involves calculating its reciprocal: \( \tan(\theta) = 1 \).

Utilizing reciprocals simplifies many trigonometric proofs and calculations. Understanding and finding reciprocals quickly is a valuable skill in solving trigonometric problems.