Is Tan Α > 1? Solve With Trig & Angle Analysis
Hey guys! Let's dive into a fun trigonometry problem today. We're tackling a question about whether the tangent of an angle (tan α) is greater than 1. To crack this, we'll analyze two statements and see if they give us enough info to answer the question. So, buckle up and let's get started!
Understanding the Core Question: When is tan α > 1?
The main question we're trying to answer is: "Is tan α > 1?" To really get what this means, let's break down the tangent function and how it behaves. Remember, in basic trigonometry, for a right-angled triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, we write it as:
tan α = Opposite / Adjacent
Now, think about when this ratio would be greater than 1. For a fraction to be greater than 1, the numerator (the opposite side in our case) needs to be longer than the denominator (the adjacent side). This has a direct connection to the angle α itself. Let's consider the unit circle, which is a super helpful tool for visualizing trigonometric functions.
In the unit circle, the tangent function increases as the angle increases from 0° to 90°. Specifically:
- When α is between 0° and 45°, the opposite side is shorter or equal to the adjacent side, so tan α ≤ 1.
- At α = 45°, the opposite and adjacent sides are equal, and tan 45° = 1.
- When α is greater than 45° but less than 90°, the opposite side is longer than the adjacent side, making tan α > 1. This is the key condition we're looking for!
So, to answer our main question, we need to figure out if the given statements can confirm whether α falls in that sweet spot between 45° and 90°. Keep this in mind as we analyze the statements. We're essentially looking for clues about the size of angle α relative to 45°.
Statement 1: α = 45° – Is It Sufficient?
Okay, let's jump into the first statement: α = 45°. At first glance, this seems pretty straightforward, right? We have a definite value for α. But let's think critically about our main question: "Is tan α > 1?"
Here’s the deal: we know that tan 45° = 1. This is a fundamental trigonometric value that you might even have memorized. If not, it's a good one to remember!
So, plugging in α = 45° into our question, we get: "Is 1 > 1?" Clearly, the answer is no. 1 is not greater than 1; it's equal to 1.
But here’s where it gets crucial for data sufficiency questions: simply getting a “no” answer doesn’t automatically mean the statement is insufficient. We need to ask ourselves: Does this statement definitively answer the question? In this case, yes, it does!
Because we know exactly what tan α is when α = 45°, we can definitively say that tan α is not greater than 1. Therefore, statement 1 alone gives us a clear, unambiguous answer. So, statement 1 is sufficient to answer the question. Awesome!
Statement 2: sin α = cos α – Decoding the Clues
Alright, let's shift our focus to the second statement: sin α = cos α. This one is a bit more interesting because it doesn't give us a direct value for α. Instead, it gives us a relationship between the sine and cosine of α. To use this, we need to understand what it implies about the angle α itself.
Remember the definitions of sine and cosine in a right-angled triangle:
- sin α = Opposite / Hypotenuse
- cos α = Adjacent / Hypotenuse
For sin α to be equal to cos α, the Opposite and Adjacent sides must be equal (since they have the same Hypotenuse). Now, think back to our earlier discussion about when tan α = 1. This happens when the opposite and adjacent sides are equal. So, the condition sin α = cos α is strongly hinting that we're dealing with a specific angle.
In fact, there's a well-known angle where the sine and cosine are equal: 45°. At 45°, both sin 45° and cos 45° are equal to √2 / 2. This is a key trigonometric identity.
So, the statement sin α = cos α implies that α = 45° (within the range of 0° to 90° for acute angles). Now we're back in familiar territory! We already analyzed what happens when α = 45° in the context of Statement 1.
Just like with Statement 1, we know that tan 45° = 1. Therefore, tan α is not greater than 1. We have a definite answer to our main question.
So, Statement 2 also provides us with enough information to answer the question definitively. This means Statement 2 is sufficient as well. You're doing great!
Putting It All Together: The Final Verdict
Okay, let's recap what we've discovered. We started with the question: "Is tan α > 1?" Then we analyzed two statements separately:
- Statement 1: α = 45° – We found that this statement is sufficient because it leads us to the conclusion that tan α is not greater than 1.
- Statement 2: sin α = cos α – This statement is also sufficient because it implies that α = 45°, which again leads us to the conclusion that tan α is not greater than 1.
Since each statement alone is sufficient to answer the question, the final answer is that either statement alone is sufficient. This type of data sufficiency question highlights the importance of understanding trigonometric relationships and being able to connect them to specific angle values.
I hope this explanation helps you guys understand the problem and the logic behind the solution. Keep practicing these types of questions, and you'll become a trigonometry pro in no time! Remember to always break down the problem, understand the definitions, and think critically about what each statement implies.
