Finding Tan 118° Given Sin 28° = P: A Trigonometry Solution

Hey guys! Let's dive into a cool trigonometry problem where we need to figure out how to express tan 118° if we know that sin 28° = p. This might seem tricky at first, but we'll break it down step by step so it's super clear. We'll explore the relationships between trigonometric functions, use some identities, and finally arrive at the answer. So, grab your thinking caps, and let's get started!

Understanding the Problem: sin 28° = p and Finding tan 118°

Okay, so the heart of this problem lies in understanding how trigonometric functions relate to each other and how angles in different quadrants behave. We're given that sin 28° = p, and our mission is to express tan 118° in terms of p. Now, 28° is an acute angle (less than 90°), while 118° is an obtuse angle (between 90° and 180°). This means we need to consider the properties of sine and tangent in different quadrants.

First off, let's think about sine. Sine is positive in both the first (0° to 90°) and second (90° to 180°) quadrants. Since 28° is in the first quadrant, sin 28° is positive, which makes sense. Now, how does this relate to 118°? Well, 118° is in the second quadrant, and we can use the fact that sin (180° - θ) = sin θ. So, sin 118° = sin (180° - 118°) = sin 62°. But wait, we know sin 28°, not sin 62°! This is where we need to think about complementary angles. Remember that cos θ = sin (90° - θ). So, cos 28° = sin (90° - 28°) = sin 62°. Aha! This connects cos 28° to our angle of interest.

Now, let's shift our focus to tangent. Tangent is positive in the first and third quadrants, but negative in the second quadrant. Since 118° is in the second quadrant, tan 118° will be negative. This is a crucial piece of information! We know that tan θ = sin θ / cos θ. So, tan 118° = sin 118° / cos 118°. We've already figured out that sin 118° is related to cos 28°, but what about cos 118°? Here, we can use the identity cos (180° - θ) = -cos θ. Thus, cos 118° = cos (180° - 62°) = -cos 62°. And since cos 62° = sin 28° = p, we have cos 118° = -p. Now we're getting somewhere!

We've established some key relationships: sin 28° = p, sin 118° = cos 28°, and cos 118° = -p. To find tan 118°, we need both sin 118° and cos 118°. We have cos 118°, but we need to express cos 28° (which is equal to sin 118°) in terms of p. This is where the Pythagorean identity comes to the rescue!

Using Trigonometric Identities: The Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, and it's super useful for problems like this. It states that sin² θ + cos² θ = 1. We can use this to relate sin 28° and cos 28°. We know sin 28° = p, so we can write:

p² + cos² 28° = 1

Now, we can solve for cos² 28°:

cos² 28° = 1 - p²

Taking the square root of both sides, we get:

cos 28° = ±√(1 - p²)

Since 28° is in the first quadrant, cosine is positive, so we take the positive square root:

cos 28° = √(1 - p²)

Remember that cos 28° = sin 118°, so we now have sin 118° expressed in terms of p: sin 118° = √(1 - p²). This is a major breakthrough! We now have both sin 118° and cos 118° in terms of p. We know sin 118° = √(1 - p²) and cos 118° = -p. Now, we can finally calculate tan 118°.

Calculating tan 118°: Putting it All Together

Okay, guys, we're in the home stretch! We know that tan θ = sin θ / cos θ. So, to find tan 118°, we simply divide sin 118° by cos 118°:

tan 118° = sin 118° / cos 118°

We've already found that sin 118° = √(1 - p²) and cos 118° = -p. Plugging these values into the equation, we get:

tan 118° = √(1 - p²) / (-p)

This can also be written as:

tan 118° = -√(1 - p²) / p

So, the value of tan 118° expressed in terms of p is -√(1 - p²) / p. You might see this written in a slightly different form, such as - (√(1 - p²)) / p, but it's the same thing. The negative sign is important because, as we discussed earlier, tangent is negative in the second quadrant.

Analyzing the Options and Choosing the Correct Answer

Now, let's take a look at the options provided and see which one matches our result. The options were:

A. √(1 - p²) / p B. p / √(1 - p²) C. √(1 + p²) / p D. p / √(1 - p²) E. √(1 - p²) / p

Notice that options A and E are very similar to our answer, but they are missing the crucial negative sign. Since tan 118° is negative, these options are incorrect. None of the other options match our result. It seems there might be a slight error in the provided options. The correct answer should be the negative of option A or E. If we were to choose the closest option, it would be either A or E, but we need to remember that the correct expression includes a negative sign.

Key Takeaways and Wrapping Up

So, what have we learned in this trigonometric adventure? We've seen how to:

• Use the relationship between sine, cosine, and tangent.
• Apply trigonometric identities like the Pythagorean identity (sin² θ + cos² θ = 1).
• Understand the signs of trigonometric functions in different quadrants.
• Express trigonometric functions of obtuse angles in terms of acute angles.

This problem highlights the importance of a solid understanding of trigonometric principles and how to apply them. It also shows that sometimes, the provided options might not be perfectly accurate, and it's crucial to understand the underlying concepts to identify the correct answer, even if it's not explicitly listed.

Great job, guys! You've tackled a challenging trigonometry problem and come out on top. Keep practicing, and you'll become trigonometric masters in no time!