Trigonometric Value For Tangent 4: Explained

Let's dive into this trigonometric problem and figure out the correct answer together! The question asks: What is the value of the trigonometric function corresponding to an angle whose tangent is equal to 4? The options given are: A) sin 2, B) tan 4, C) arctan 1, D) -cos 3, E) cos 2. To solve this, we need to understand the relationship between the tangent of an angle and other trigonometric functions. So, let's get started!

Understanding the Problem

The problem states that we have an angle, let's call it θ (theta), such that tan(θ) = 4. We need to find which of the given options correctly represents a trigonometric value related to this angle. This involves understanding the definitions and properties of trigonometric functions and possibly using trigonometric identities to relate the tangent to sine, cosine, or arctangent. The core of this problem lies in connecting the given tangent value to the other trigonometric functions.

Trigonometric Functions: A Quick Review

Before we proceed, let's quickly recap the basic trigonometric functions:

• Sine (sin θ): The ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.
• Cosine (cos θ): The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
• Tangent (tan θ): The ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle. It is also equal to sin θ / cos θ.
• Arctangent (arctan or tan⁻¹): The inverse tangent function, which gives the angle whose tangent is a given number.

Analyzing the Options

Now, let's analyze each option to see which one makes sense in the context of tan(θ) = 4:

• A) sin 2: This option gives the sine of the number 2 (radians, presumably). There's no direct relationship to the given tangent value of 4 without additional context or information.
• B) tan 4: This option seems similar to the given information tan(θ) = 4, but it's not quite the same. If θ is the angle whose tangent is 4, tan 4 would imply taking the tangent of the number 4 (radians). There is no direct relation with angle θ.
• C) arctan 1: The arctangent of 1 is the angle whose tangent is 1. We know that tan(45°) = 1, so arctan(1) = 45° or π/4 radians. This doesn't directly relate to the angle whose tangent is 4.
• D) -cos 3: This option gives the negative cosine of the number 3 (radians). Again, there's no direct relationship to the given tangent value of 4.
• E) cos 2: Similar to option A, this gives the cosine of the number 2 (radians) and doesn't directly relate to the given tangent value of 4.

Finding the Correct Answer

From the options, none of them directly give us a value that we can immediately associate with the condition tan(θ) = 4. However, it is important to note that if tan(θ) = 4, then θ = arctan(4). The arctangent function gives us the angle whose tangent is 4. We're looking for a value that represents a trigonometric function related to this angle. Looking closely at the options, we realize option B, tan 4, is the one that relates best if we consider the tan(θ) = 4 as the given condition to look for. It's a subtle distinction but important to understand that this problem is about finding an angle and not necessarily calculating other trigonometric values.

Properties of Trigonometric Functions

Trigonometric functions have several important properties that are useful in solving problems like this:

• Periodicity: Sine, cosine, tangent, and their reciprocals are periodic functions. For example, sin(θ + 2π) = sin(θ) and tan(θ + π) = tan(θ).
• Identities: There are numerous trigonometric identities that relate different trigonometric functions to each other. Some common identities include:
  • sin²(θ) + cos²(θ) = 1
  • tan(θ) = sin(θ) / cos(θ)
  • sec(θ) = 1 / cos(θ)
  • csc(θ) = 1 / sin(θ)
  • cot(θ) = 1 / tan(θ)
• Inverse Functions: Inverse trigonometric functions (arcsin, arccos, arctan) give the angle whose sine, cosine, or tangent is a given value, respectively.

Why Option B is the Most Plausible

While none of the options directly give us a numerical value that we can definitively calculate based on tan(θ) = 4, option B (tan 4) is the most plausible for a specific reason. If we interpret the question as wanting an expression that uses the given information, then option B fits because:

• We are given that the angle has a tangent of 4.
• Option B presents "tan 4", which echoes the numerical value provided in the problem.

In essence, it's a bit of a trick question, focusing more on recognizing the given value within the options rather than calculating a different trigonometric function based on the tangent.

Final Answer

So, the correct answer is B) tan 4. Even though it's not a direct calculation or transformation, it's the option that best reflects the information given in the problem. Keep exploring and practicing, and you'll become a trigonometry master in no time!