Solving Trigonometry Problems: Finding Correct Statements

Hey guys! Let's dive into a fun trigonometry problem. We're given that $\sin 40^\circ = \frac{1}{p}$, and our mission is to figure out which of the following statements are true. Don't worry; it's not as scary as it sounds. We'll break it down step by step. This problem involves understanding trigonometric identities and how angles relate to each other. So, let's get started and see what we can uncover together! The main goal is to analyze each statement and use our trigonometric knowledge to determine its validity. It's like a puzzle, and we are the detectives! So let's put on our thinking caps and get ready to solve this. Ready?

Understanding the Given Information

First things first, let's understand what we're starting with. We know that $\sin 40^\circ = \frac{1}{p}$. This means that the sine of a 40-degree angle is equal to $\frac{1}{p}$. Remember, sine is a fundamental trigonometric function that relates an angle in a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The value of $p$ is essentially a scaling factor here. We're not given the exact value of $p$, but it's important to know that since the sine value is positive, $p$ must also be positive (as $\frac{1}{p}$ is positive). This basic information is our launchpad for solving the rest of the problem. It's important to keep this in mind as we proceed. Think of it as the first clue in our investigation. Now, we'll look at how the given sine value can help us find other trigonometric values.

What is Sine?

Before we move on, let's quickly recap what sine is all about. Sine, in a right-angled triangle, is the ratio of the length of the side opposite the angle to the length of the hypotenuse. This is often remembered by the acronym SOH (Sine = Opposite / Hypotenuse). So, in our case, if we visualize a right-angled triangle where one of the acute angles is 40 degrees, the ratio of the opposite side to the hypotenuse is $\frac{1}{p}$. This simple concept is crucial for understanding and solving the problem. Keeping this definition clear in our minds will allow us to correctly analyze each option presented in the problem. This basic understanding of trigonometric functions is extremely important, so make sure that you remember it.

The Significance of $p$

The value of $p$ holds a key role in the trigonometric context. As $\sin 40^\circ = \frac{1}{p}$, it's essential to recognize that $p$ is the reciprocal of $\sin 40^\circ$. This implies that $p$ is greater than 1, because the sine of an angle, as we know, has a value between -1 and 1. This is a crucial point to consider when we're evaluating the other options. It gives us a reference to how we can relate different trigonometric values. Essentially, knowing the relationship between $\sin 40^\circ$ and $p$ helps us solve for other unknown values. Understanding $p$ helps in understanding all the other trigonometric functions. It's useful to know that $p$ is a positive value greater than 1. This knowledge is very important as it gives us a clue to how the different trigonometric values are related to each other.

Analyzing the Statements

Now, let's get to the main course: the statements! We need to examine each one and see if it's true based on what we know about $\sin 40^\circ$ and trigonometric identities. This is where the fun begins. We'll use our knowledge of trigonometric identities to determine the validity of each statement. Remember, we're looking for statements that are true, so it's important to be precise and methodical in our approach. Each statement will test a different aspect of trigonometric knowledge, which means you will have to remember multiple trigonometric functions in order to proceed. Let's begin with statement A.

Statement A: $\sin 140^\circ = \frac{1}{p}$

Here, we need to find out if the sine of 140 degrees is also $\frac{1}{p}$. To do this, we'll use the fact that $\sin (180^\circ - x) = \sin x$. This is a super important identity, guys! It means that the sine of an angle is equal to the sine of its supplement (the angle that adds up to 180 degrees). Since 140 degrees is the supplement of 40 degrees (180 - 40 = 140), we can say that $\sin 140^\circ = \sin (180^\circ - 40^\circ) = \sin 40^\circ$. And since we know $\sin 40^\circ = \frac{1}{p}$, it follows that $\sin 140^\circ = \frac{1}{p}$. So, statement A is TRUE! We successfully used a trigonometric identity and our initial information to determine the truth of this statement. It's a clear indication of how angles and trigonometric functions are interlinked. Remember that $\sin (180^\circ - x) = \sin x$. This is a key concept to remember. It's like a secret code to unlock the solution.

Statement B: $\cos 230^\circ = -\frac{\sqrt{p^2 - 1}}{p}$

Okay, let's tackle statement B. We have to determine if the cosine of 230 degrees is equal to $- \frac{\sqrt{p^2 - 1}}{p}$. This requires a few more steps. First, let's relate 230 degrees to a more familiar angle. 230 degrees is in the third quadrant, so we can express it as 180 + 50 degrees. Remember, cosine is negative in the third quadrant. We can use the identity: $\cos(180^\circ + x) = -\cos x$. Thus, $\cos 230^\circ = \cos(180^\circ + 50^\circ) = -\cos 50^\circ$. Now, we need to find $\cos 50^\circ$. We know that $\sin 40^\circ = \frac{1}{p}$, and we can use the complementary angle identity, $\sin x = \cos(90^\circ - x)$. Thus, $\sin 40^\circ = \cos(90^\circ - 40^\circ) = \cos 50^\circ$. Therefore, $\cos 50^\circ = \frac{1}{p}$. Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, we can write $\cos 50^\circ$ as $\sqrt{1 - \sin^2 50^\circ}$. Since $\sin 50^\circ = \cos 40^\circ$, we need to find $\cos 40^\circ$. We know $\sin^2 40^\circ + \cos^2 40^\circ = 1$, so $\cos 40^\circ = \sqrt{1 - \sin^2 40^\circ} = \sqrt{1 - (\frac{1}{p})^2} = \sqrt{1 - \frac{1}{p^2}} = \frac{\sqrt{p^2 - 1}}{p}$. Hence, $\cos 50^\circ = \frac{\sqrt{p^2 - 1}}{p}$. Therefore, $\cos 230^\circ = -\cos 50^\circ = -\frac{\sqrt{p^2 - 1}}{p}$. So, statement B is also TRUE! It's all about piecing together different trigonometric identities and relationships. It shows how a sequence of mathematical steps can confirm a complex trigonometric expression. Remember these identities, guys; they are super useful!

Statement C: Finding the Value of $\tan 320^\circ$

Now let's analyze statement C. We must determine the value of $\tan 320^\circ$. The 320 degrees lies in the fourth quadrant, so we can express it as 360 - 40 degrees. The tangent function is negative in the fourth quadrant. Thus, $\tan 320^\circ = \tan (360^\circ - 40^\circ) = -\tan 40^\circ$. We know that $\tan x = \frac{\sin x}{\cos x}$. Using the known information we can now solve it. As we know $\sin 40^\circ = \frac{1}{p}$ and $\cos 40^\circ = \frac{\sqrt{p^2 - 1}}{p}$, then $\tan 40^\circ = \frac{\sin 40^\circ}{\cos 40^\circ} = \frac{\frac{1}{p}}{\frac{\sqrt{p^2 - 1}}{p}} = \frac{1}{\sqrt{p^2 - 1}}$. Therefore, $\tan 320^\circ = -\tan 40^\circ = -\frac{1}{\sqrt{p^2 - 1}}$.

Conclusion

Alright! We've successfully analyzed all the statements. Let's recap. Statement A, which stated that $\sin 140^\circ = \frac{1}{p}$, is true. Statement B, which claimed $\cos 230^\circ = -\frac{\sqrt{p^2 - 1}}{p}$, is also true. And for statement C, we found that $\tan 320^\circ = -\frac{1}{\sqrt{p^2 - 1}}$.

So, the correct statements are A and B, as they are the only ones we found to be true given the information provided! Isn't that awesome, guys? We successfully navigated through a trigonometry problem by understanding the basics of sine, cosine, and tangent and applying trigonometric identities. The key is to break down the problem into smaller parts, identify the relevant formulas, and apply them step by step. Keep practicing, and you'll become a trigonometry master in no time!