Trigonometry Values In Quadrant III Analyzing Tan Θ And Csc Θ

Trigonometry, the study of relationships between angles and sides of triangles, often presents intriguing puzzles. One such puzzle arises when we're given specific values for trigonometric functions and asked to determine the quadrant in which the angle lies. Let's dissect the statement: "$\tan \theta = -\frac{12}{5}$, $\csc \theta = -\frac{13}{5}$, and the terminal point determined by $\theta$ is in quadrant 3." Our mission is to determine if this statement holds true or if there's a contradiction lurking within.

Quadrant III holds a special place in the trigonometric world. It's where both the x and y coordinates are negative. This negativity dictates the signs of various trigonometric functions. Understanding the signs of trigonometric functions in each quadrant is crucial for solving these types of problems. In Quadrant III, tangent ($\tan$) is positive (since it's the ratio of y to x, both negative), while sine ($\sin$) and cosine ($\cos$) are negative. Their reciprocals, cosecant ($\csc$) and secant ($\ ext{sec}$), are also negative. This foundational knowledge sets the stage for our analysis. We'll delve deeper into how these sign conventions play a pivotal role in validating trigonometric statements.

Now, let's break down the given information. We're told that $\tan \theta = -\frac{12}{5}$. This immediately raises a red flag. As we established, $\tan \theta$ should be positive in Quadrant III. A negative tangent value contradicts the quadrant specification. This single piece of information hints at the statement's potential falsehood. However, we can't jump to conclusions just yet. We need to examine the other given value, $\csc \theta = -\frac{13}{5}$, to see if it aligns with the quadrant. This is the step-by-step method to solve it and in this step-by-step method, there is a logical explanation about why the statement is invalid.

The next step is to examine the cosecant value. $\csc \theta$, being the reciprocal of $\sin \theta$, should indeed be negative in Quadrant III. The given value of $-\frac{13}{5}$ seems to align with our quadrant rules. However, the contradiction with the tangent value remains. We have conflicting information: a negative tangent indicating a quadrant where tangent should be positive. This conflict is the key to unraveling the puzzle. We will use trigonometric identities to confirm the truthfulness of this conflict and to further confirm this conflict. The usage of trigonometric identities is important because this can confirm the truth value of the statement.

To definitively resolve this trigonometric conundrum, we must reconcile the conflicting information about the tangent and cosecant values within the context of Quadrant III. The critical observation is the sign of the tangent function. In Quadrant III, both the x and y coordinates are negative. Therefore, the tangent function, defined as the ratio of y to x, must be positive (negative divided by negative equals positive). The given value of $\tan \theta = -\frac{12}{5}$ directly contradicts this fundamental property of Quadrant III.

Let's solidify this understanding with a visual analogy. Imagine a point in Quadrant III. Draw a line from the origin to this point. This line forms the hypotenuse of a right triangle. The x and y components of the point represent the adjacent and opposite sides of the triangle, respectively. Since both x and y are negative, their ratio (which defines the tangent) must be positive. This geometric interpretation reinforces the incompatibility of a negative tangent in Quadrant III. Furthermore, let's use the trigonometric identity $\tan^2(\theta) + 1 = \sec^2(\theta)$ to further validate the tangent value. The value of $\tan^2(\theta)$ should always be positive since it is a square. Therefore, the left-hand side of the trigonometric identity $\tan^2(\theta) + 1$ will always be greater than 1. This trigonometric identity further ensures that the tangent function in quadrant III is a positive tangent function.

The fact that $\csc \theta = -\frac{13}{5}$ doesn't negate the contradiction. While a negative cosecant is consistent with Quadrant III (where sine, its reciprocal, is negative), it cannot override the impossible scenario of a negative tangent. The tangent's sign is the deciding factor in this case. This is because the tangent function value that was provided contradicts with the rule of tangent value in the third quadrant, where the tangent value should be positive instead of negative. The negative tangent value therefore determines whether the statement holds or not.

To further illustrate, let's attempt to construct a right triangle based on the given information. If $\tan \theta = -\frac{12}{5}$, we might initially think of a triangle with opposite side -12 and adjacent side 5 (or vice versa). However, this would place the angle in either Quadrant II (where sine is positive and cosine is negative) or Quadrant IV (where cosine is positive and sine is negative), not Quadrant III. The cosecant value, $\csc \theta = -\frac{13}{5}$, reinforces that sine is negative, which is consistent with Quadrants III and IV. But again, the tangent value creates an insurmountable obstacle. This thought experiment highlights the interconnectedness of trigonometric functions and quadrant rules.

Now that we've thoroughly analyzed the trigonometric values and their implications within Quadrant III, let's address the original statement and the options provided. The statement presents a scenario with a negative tangent and a negative cosecant in Quadrant III. Our analysis has revealed that this is fundamentally impossible due to the sign of the tangent function in Quadrant III.

Let's consider the provided options (although only two are given in the prompt, we can infer a general structure): A. cannot be true because $\tan \theta$ must be less than 1. B. cannot be true. The crucial point is the reasoning behind why the statement cannot be true. Option A's reasoning – that $\tan \theta$ must be less than 1 – is incorrect. Tangent can take on values greater than 1. The actual reason is the sign contradiction. The correct explanation, therefore, aligns with Option B, but with a more precise justification. Option B is more accurate because it doesn't impose any numerical limit on $\tan \theta$, it simply states that the statement cannot be true. However, we can enhance the explanation to specify why it cannot be true.

Therefore, the statement "$\tan \theta = -\frac{12}{5}$, $\csc \theta = -\frac{13}{5}$, and the terminal point determined by $\theta$ is in quadrant 3" cannot be true because the given value of $\tan \theta$ is negative, which contradicts the fact that $\tan \theta$ must be positive in Quadrant III. This conclusion is reached by applying the fundamental principles of trigonometry and quadrant analysis. The contradiction stems from the inherent properties of trigonometric functions within specific quadrants. The tangent function, defined as the ratio of the sine to the cosine, is positive in Quadrant III because both sine and cosine are negative in this quadrant. The negative value provided for the tangent directly clashes with this established rule. Thus, the statement is demonstrably false. To further ensure that the correctness of the solution, other trigonometric identities could have been used to solve this problem and to achieve the same result. This is a method to ensure the correctness of the result of a trigonometric problem.

In summary, the statement is false due to the sign of the tangent function in Quadrant III. This exercise underscores the importance of understanding the sign conventions of trigonometric functions in different quadrants and how these signs dictate the possible values of these functions. The ability to identify such contradictions is a key skill in trigonometry and a testament to the interconnectedness of trigonometric concepts.