Convert Degrees To Radians & Simplify Trig Expressions

Hey guys! Today, we're diving into some cool math problems involving converting degrees to radians and simplifying trigonometric expressions. These are fundamental concepts in trigonometry, and mastering them will definitely boost your math skills. Let's break it down step by step, making sure everyone understands the logic and methods behind each solution. So, grab your calculators, and let’s get started!

Converting Degrees to Radians

Okay, so our first task is to convert 330 degrees into radians. You might be wondering, what exactly are radians, and why do we need them? Radians are another way to measure angles, just like degrees. Think of it this way: degrees divide a circle into 360 parts, while radians relate the angle to the radius of the circle. One full circle is 2π radians, which is equivalent to 360 degrees. This relationship is key to our conversion.

The Conversion Formula

The golden rule for converting degrees to radians is using the conversion factor:

radians = degrees × (π / 180)

This formula is derived from the fact that π radians equals 180 degrees. So, if we want to find out how many radians are in 330 degrees, we just plug the value into our formula.

Applying the Formula to 330 Degrees

Let’s calculate: radians = 330 × (π / 180).

First, simplify the fraction 330/180. Both numbers are divisible by 30, so we can simplify it to 11/6.

Now, our equation looks like this: radians = (11/6)π

So, 330 degrees is equal to (11/6)π radians. Looking at our options, we need to see which one matches this value.

Analyzing the Options

We have the following options:

A) 756.30 rad B) 378.15 rad C) 1.833 rad D) 1.833π rad

Option D, 1.833π rad, seems promising because it includes π, just like our calculated answer. To confirm, let's convert (11/6)π to a decimal to see if it matches 1.833π. When you divide 11 by 6, you get approximately 1.833. Therefore, (11/6)π is indeed equal to 1.833π.

Why This Matters

Understanding how to convert between degrees and radians is crucial in many areas of math and physics. Radians are especially important in calculus and other advanced math topics because they simplify many formulas and equations. Plus, many scientific calculators use radians as the default angle unit, so knowing how to work with them is super practical.

So, the correct answer is D) 1.833π rad. You nailed it!

Simplifying Trigonometric Expressions

Next up, let's tackle simplifying trigonometric expressions using trigonometric identities. This might sound intimidating, but it's like solving a puzzle. We use known relationships between trigonometric functions to make complex expressions simpler. The expression we need to simplify is:

(1 - cos² θ) / tan² θ

Understanding Trigonometric Identities

Before we jump into the simplification, let's quickly review some fundamental trigonometric identities. These are the building blocks we'll use to solve this problem.

1. Pythagorean Identity: sin² θ + cos² θ = 1. This is probably the most famous one, and it's super useful.
2. Definition of Tangent: tan θ = sin θ / cos θ
3. Other forms of Pythagorean Identity: We can rearrange the Pythagorean identity to get other useful forms, like sin² θ = 1 - cos² θ and cos² θ = 1 - sin² θ.

Step-by-Step Simplification

Now, let's simplify our expression step by step:

1. Recognize the Pythagorean Identity: Look at the numerator (1 - cos² θ). Does it ring a bell? It's actually a direct match for one of the forms of the Pythagorean identity. We know that sin² θ = 1 - cos² θ. So, we can replace (1 - cos² θ) with sin² θ. Our expression now looks like this:

   sin² θ / tan² θ

2. Rewrite Tangent: Next, let's deal with the tan² θ in the denominator. We know that tan θ = sin θ / cos θ. Therefore, tan² θ = (sin² θ / cos² θ). Substitute this into our expression:

   sin² θ / (sin² θ / cos² θ)

3. Divide by a Fraction: Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite our expression as:

   sin² θ × (cos² θ / sin² θ)

4. Simplify: Now, we can cancel out the sin² θ terms in the numerator and the denominator:

   (sin² θ × cos² θ) / sin² θ = cos² θ

So, the simplified form of our expression is cos² θ.

Matching with the Options

Let's look at our options:

A) cos² θ B) -cos² θ C) sec² θ D) -sec² θ

Our simplified expression, cos² θ, perfectly matches option A. So, we've found our answer!

Why Simplifying Matters

Simplifying trigonometric expressions isn't just a math exercise; it's a skill that's invaluable in higher-level mathematics, physics, and engineering. Simplified expressions are easier to work with, making complex problems more manageable. Plus, it shows a deep understanding of the relationships between trigonometric functions, which is always a win!

Therefore, the correct answer is A) cos² θ. Great job!

Final Thoughts

So, guys, we've successfully converted degrees to radians and simplified a trigonometric expression. Remember, the key to mastering these concepts is practice. The more you work with these formulas and identities, the more comfortable you'll become. Don't be afraid to make mistakes – that's how we learn! Keep practicing, and you'll become a math whiz in no time. Keep up the awesome work, and I'll catch you in the next math adventure!