Finding Cot Θ When Sec Θ Is -5/4 And Θ Is In Quadrant II
Hey guys! Let's dive into a fun math problem today. We're going to figure out the value of cot θ (cotangent theta) when we know that sec θ (secant theta) is -5/4, and θ (theta) is chilling in the second quadrant. Don't worry, it sounds trickier than it is! We'll break it down step by step so it's super easy to follow.
Understanding the Problem
So, the heart of the problem lies in understanding trigonometric functions and their relationships, especially in different quadrants of the unit circle. We're given that sec θ = -5/4 and that θ lies in Quadrant II. This is super important because the quadrant tells us about the signs (+ or -) of the trigonometric functions. Remember, each quadrant has specific trigonometric functions that are positive or negative, and knowing this is key to solving the problem correctly. Secant (sec) is the reciprocal of cosine (cos), so sec θ = 1/cos θ. This means we can find cos θ easily. Also, cotangent (cot) is the reciprocal of tangent (tan), and cot θ = 1/tan θ. Tangent, in turn, is sin θ / cos θ, so cot θ can also be expressed as cos θ / sin θ. The fact that θ is in Quadrant II is crucial. In this quadrant, sine (sin θ) is positive, while cosine (cos θ) and tangent (tan θ) are negative. This will help us determine the correct sign for our final answer. We will use trigonometric identities and the properties of quadrants to find the value of cot θ. It is essential to remember the definitions of trigonometric functions in terms of the sides of a right-angled triangle (SOH CAH TOA) and their signs in different quadrants. By carefully applying these concepts, we can accurately determine the value of cot θ.
Step-by-Step Solution
1. Find cos θ
We know that sec θ = -5/4, and since sec θ is the reciprocal of cos θ, we can say:
cos θ = 1 / sec θ = 1 / (-5/4) = -4/5
So, we've found that cos θ = -4/5. Easy peasy, right?
2. Find sin θ
Now, we need to find sin θ. We can use the Pythagorean identity, which is a fundamental relationship in trigonometry:
sin² θ + cos² θ = 1
We already know cos θ, so we can plug that in:
sin² θ + (-4/5)² = 1
sin² θ + 16/25 = 1
Now, let's isolate sin² θ:
sin² θ = 1 - 16/25
sin² θ = 9/25
To find sin θ, we take the square root of both sides:
sin θ = ±√(9/25)
sin θ = ±3/5
But hold on! Remember that θ is in Quadrant II, where sin θ is positive. So, we choose the positive value:
sin θ = 3/5
Great! We've got sin θ.
3. Find cot θ
We know that cot θ = cos θ / sin θ. We've already found both cos θ and sin θ, so let's plug them in:
cot θ = (-4/5) / (3/5)
To divide fractions, we multiply by the reciprocal:
cot θ = (-4/5) * (5/3)
cot θ = -4/3
And there you have it! We've found that cot θ = -4/3.
Why the Quadrant Matters
Okay, let's talk a bit more about why the quadrant information is so important. Trigonometric functions change signs depending on which quadrant the angle falls into. Think of it like a coordinate plane:
- Quadrant I (0° - 90°): All trigonometric functions (sin, cos, tan, csc, sec, cot) are positive.
- Quadrant II (90° - 180°): Only sine (sin) and its reciprocal, cosecant (csc), are positive. Cosine (cos), tangent (tan), and their reciprocals are negative.
- Quadrant III (180° - 270°): Only tangent (tan) and its reciprocal, cotangent (cot), are positive. Sine (sin), cosine (cos), and their reciprocals are negative.
- Quadrant IV (270° - 360°): Only cosine (cos) and its reciprocal, secant (sec), are positive. Sine (sin), tangent (tan), and their reciprocals are negative.
A helpful mnemonic to remember this is "All Students Take Calculus" (ASTC), which tells you which functions are positive in each quadrant. In our problem, knowing that θ is in Quadrant II helped us determine that sin θ should be positive and cos θ should be negative. If we ignored this information, we might have ended up with the wrong sign for our answer. So, always pay attention to the quadrant!
Common Mistakes to Avoid
Trigonometry can be a bit tricky, so let's talk about some common mistakes people make when solving these types of problems, so you can steer clear of them:
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Forgetting the Quadrant: This is the biggest one! As we discussed, the quadrant dictates the signs of the trigonometric functions. Always, always, always consider the quadrant when determining the sign of your answer. If you forget this, you're likely to get the wrong answer.
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Mixing Up Reciprocal Identities: It's easy to mix up which function is the reciprocal of which. Remember: sec θ = 1/cos θ, csc θ = 1/sin θ, and cot θ = 1/tan θ. Double-check your identities to make sure you're using the correct ones. A small mistake here can throw off your entire calculation.
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Incorrectly Applying the Pythagorean Identity: The Pythagorean identity (sin² θ + cos² θ = 1) is super useful, but make sure you're plugging in the values correctly. Don't forget to square the trigonometric functions! Also, be careful when solving for sin θ or cos θ – remember to take the square root and consider both positive and negative possibilities before using the quadrant information to decide on the correct sign.
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Math Errors: Simple arithmetic errors can happen, especially when dealing with fractions and square roots. Take your time, double-check your calculations, and don't rush through the steps. It's better to be accurate than fast!
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Not Simplifying Completely: Make sure you simplify your final answer as much as possible. For example, if you end up with a fraction that can be reduced, simplify it to its lowest terms. A non-simplified answer might not be marked as wrong, but it's always good practice to present your answer in the simplest form.
By being aware of these common mistakes, you can increase your chances of solving trigonometry problems correctly and confidently.
Alternative Approaches
While we solved this problem using the Pythagorean identity and the definitions of trigonometric functions, there are other ways to approach it. Here's a quick look at an alternative method:
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Using a Right Triangle: You can visualize the problem using a right triangle. Since sec θ = -5/4, and sec θ is hypotenuse/adjacent, we can think of a right triangle where the hypotenuse has a length of 5 and the adjacent side has a length of -4 (remember, cosine is negative in Quadrant II). Then, we can use the Pythagorean theorem to find the length of the opposite side:
a² + b² = c²
(-4)² + b² = 5²
16 + b² = 25
b² = 9
b = ±3
Since we're in Quadrant II, sine is positive, so we take b = 3.
Now we have all three sides of the triangle. We know that cot θ = adjacent/opposite, so:
cot θ = -4/3
This method provides a visual way to understand the relationships between the trigonometric functions and the sides of a right triangle. It can be particularly helpful for students who are more visual learners. However, it's important to remember that the lengths of the sides of the triangle can be positive or negative depending on the quadrant, so you still need to consider the quadrant information carefully.
Conclusion
So, after all that, we've successfully found that cot θ = -4/3 when sec θ = -5/4 and θ is in Quadrant II. We did it by using the reciprocal identity, the Pythagorean identity, and a little bit of quadrant knowledge. Remember, the key is to break the problem down into smaller steps and understand the relationships between the trigonometric functions. Keep practicing, and you'll become a trigonometry whiz in no time!
I hope this explanation was helpful and clear. If you have any questions or want to tackle another problem, just let me know. Happy math-ing, guys!