Calculate Sin 45° / Cos 30° Using Trig Formulas

Oct 13, 2025 by ADMIN 48 views

Calculate sin 45° / cos 30° Using Trigonometric Formulas

Hey guys! Today, we're diving into a classic trigonometry problem: calculating the value of sin 45° divided by cos 30°. It might sound intimidating, but trust me, it's totally manageable once we break it down. We'll use our knowledge of trigonometric formulas and special angles to solve this step-by-step. So, grab your calculators (or your unit circles!), and let's get started!

Understanding Trigonometric Formulas

Before we jump into the calculation, let's quickly recap the trigonometric formulas we'll be using. These formulas are the foundation of trigonometry and help us relate angles and sides in right triangles. Understanding these relationships is super crucial for solving trigonometric problems. The main players here are sine (sin), cosine (cos), and tangent (tan). Remember SOH CAH TOA? It's a handy mnemonic to remember the definitions:

SOH: Sine = Opposite / Hypotenuse
CAH: Cosine = Adjacent / Hypotenuse
TOA: Tangent = Opposite / Adjacent

For this specific problem, we'll primarily focus on sine and cosine. Sine gives us the ratio of the opposite side to the hypotenuse, while cosine gives us the ratio of the adjacent side to the hypotenuse. Now, these ratios become especially interesting when we deal with special angles like 30°, 45°, and 60°. These angles pop up frequently in trigonometry, and knowing their sine, cosine, and tangent values can save us a lot of time and effort. We often use special triangles, like the 45-45-90 and 30-60-90 triangles, to figure out these values.

The 45-45-90 triangle is an isosceles right triangle, meaning it has two equal angles (45°) and two equal sides. If we let the length of the two equal sides be 1, then by the Pythagorean theorem, the hypotenuse will be √2. This allows us to easily find sin 45° and cos 45°.

On the other hand, the 30-60-90 triangle has angles of 30°, 60°, and 90°. If we let the side opposite the 30° angle be 1, then the hypotenuse is 2, and the side opposite the 60° angle is √3. This triangle helps us determine the sine and cosine of 30° and 60°.

So, keep these formulas and special triangles in mind as we move forward. They're the key to unlocking the solution to our problem. Understanding these basics makes tackling more complex problems way less scary, trust me! We're building a solid foundation here, and that's what matters most.

Determining sin 45°

Okay, let's dive into finding the value of sin 45°. As we talked about earlier, 45° is a special angle, which means we can use the properties of a 45-45-90 triangle to figure this out. Imagine a right triangle where two of the angles are 45 degrees. This automatically means the third angle is 90 degrees (because the angles in a triangle always add up to 180°). What's cool about this type of triangle is that the two sides opposite the 45-degree angles are equal in length. Let's make things simple and say each of those sides has a length of 1.

Now, to find the hypotenuse (the side opposite the 90-degree angle), we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our case, that looks like this: hypotenuse² = 1² + 1² = 2. So, the hypotenuse is the square root of 2 (√2).

Remember our SOH CAH TOA? To find the sine of an angle, we use SOH, which means Sine = Opposite / Hypotenuse. For sin 45°, the side opposite the 45-degree angle has a length of 1, and the hypotenuse has a length of √2. So, sin 45° = 1 / √2. But we're not quite done yet! It's common practice to rationalize the denominator, which means we don't want a square root in the bottom of our fraction.

To do this, we multiply both the numerator (the top part of the fraction) and the denominator (the bottom part) by √2. This gives us: sin 45° = (1 * √2) / (√2 * √2) = √2 / 2. And there you have it! The value of sin 45° is √2 / 2. Knowing this value is super helpful in tons of trigonometry problems, so it's a good one to have in your mental toolbox. We've tackled the first part of our problem, and now we're ready to move on to finding cos 30°.

Determining cos 30°

Alright, let's shift our focus to finding the value of cos 30°. Just like 45°, 30° is another one of those special angles that pops up all the time in trigonometry. To figure out cos 30°, we're going to use our knowledge of the 30-60-90 triangle. This triangle has angles of 30 degrees, 60 degrees, and 90 degrees, and its sides have a special relationship that makes finding trigonometric values a breeze.

Imagine a 30-60-90 triangle. The side lengths in this triangle are always in a specific ratio. If we let the side opposite the 30-degree angle have a length of 1, then the hypotenuse (the side opposite the 90-degree angle) has a length of 2, and the side opposite the 60-degree angle has a length of √3. This ratio is key to finding our cosine value.

Now, let's bring back our SOH CAH TOA mnemonic. To find the cosine of an angle, we use CAH, which stands for Cosine = Adjacent / Hypotenuse. In our 30-60-90 triangle, the side adjacent to the 30-degree angle (that's the side next to it, but not the hypotenuse) has a length of √3, and the hypotenuse has a length of 2. So, cos 30° = √3 / 2.

And that's it! We've found that the value of cos 30° is √3 / 2. This is another value that's super useful to remember, as it'll come up again and again in your trigonometry adventures. We've now successfully determined both sin 45° and cos 30°, which means we're ready to tackle the final step: dividing them!

Calculating sin 45° / cos 30°

Okay, we've done the groundwork, and now it's time for the grand finale! We know that sin 45° = √2 / 2 and cos 30° = √3 / 2. Our mission is to calculate sin 45° / cos 30°, which means we're dividing one fraction by another. Remember how to do that? Dividing by a fraction is the same as multiplying by its reciprocal (flipping it upside down).

So, sin 45° / cos 30° is the same as (√2 / 2) / (√3 / 2). To make things easier, we'll rewrite this as a multiplication problem: (√2 / 2) * (2 / √3). Notice how we flipped the second fraction (√3 / 2) to its reciprocal (2 / √3).

Now, we can multiply the numerators together and the denominators together: (√2 * 2) / (2 * √3) = 2√2 / 2√3. We've got a 2 in both the numerator and the denominator, so we can cancel those out, leaving us with √2 / √3. But just like before, we don't want a square root in the denominator. Time to rationalize again!

To rationalize the denominator, we multiply both the numerator and the denominator by √3: (√2 * √3) / (√3 * √3) = √6 / 3. And there we have it! The final answer to our problem is √6 / 3. We've successfully calculated sin 45° / cos 30° using our knowledge of trigonometric formulas and special angles.

Conclusion

Guys, we did it! We successfully calculated sin 45° / cos 30° using trigonometric formulas. We started by understanding the basic trigonometric ratios (SOH CAH TOA) and then focused on the special angles of 45° and 30°. We used the properties of 45-45-90 and 30-60-90 triangles to determine the values of sin 45° and cos 30°. Finally, we divided those values, rationalized the denominator, and arrived at our answer: √6 / 3.

This problem is a great example of how trigonometry combines geometry and algebra. By understanding the relationships between angles and sides in triangles, we can solve a wide variety of problems. Don't be intimidated by trigonometric functions – break them down step by step, and you'll find they're much more manageable than they seem. Keep practicing, and you'll become a trigonometry whiz in no time! Remember, the key is to understand the fundamental concepts and apply them methodically. You've got this! Now go tackle some more trig problems and show them who's boss!