Equivalent Expression For Cos(40°)cos(10°) + Sin(40°)sin(10°)

Hey guys! Today, we're diving deep into the fascinating world of trigonometry, specifically tackling an expression that might look a bit intimidating at first glance: ${\cos(40^\circ)\cos(10^\circ) + \sin(40^\circ)\sin(10^\circ)}$. But don't worry, we'll break it down together and uncover the elegant solution hidden within. This isn't just about finding the answer; it's about understanding the underlying principles that govern trigonometric functions. So, buckle up and let's embark on this mathematical adventure!

Decoding the Expression: Spotting the Pattern

At the heart of this problem lies a fundamental concept in trigonometry: trigonometric identities. These identities are like secret codes that allow us to rewrite expressions in different but equivalent forms. The expression ${\cos(40^\circ)\cos(10^\circ) + \sin(40^\circ)\sin(10^\circ)}$ strongly hints at one such identity, specifically the cosine angle addition and subtraction formulas. Recognizing these patterns is the first crucial step in solving trigonometric problems. Think of it as detective work, where you're looking for clues that lead you to the solution. This particular expression looks remarkably similar to the cosine angle addition or subtraction formula, which is our key to unlocking this puzzle.

Now, let's delve a bit deeper into these formulas. The cosine angle subtraction formula states that ${\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)}$. Notice the striking resemblance between this formula and our given expression. If we can successfully map our expression onto this formula, we'll be well on our way to finding the equivalent form. The ability to recognize these patterns and apply the correct identities is a hallmark of strong trigonometric understanding. It's not just about memorizing formulas; it's about understanding how they work and when to use them.

The Cosine Angle Subtraction Formula: Our Guiding Light

Let's put the cosine angle subtraction formula under the spotlight: ${\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)}$. This powerful identity is the cornerstone of our solution. By carefully comparing this formula with our expression, ${\cos(40^\circ)\cos(10^\circ) + \sin(40^\circ)\sin(10^\circ)}$, we can start to see the pieces of the puzzle falling into place. The elegance of mathematics often lies in recognizing these connections and using them to simplify complex problems. It's like having a secret decoder ring that allows you to translate one form into another.

If we let ${A = 40^\circ}$ and ${B = 10^\circ}$, we can directly substitute these values into the cosine angle subtraction formula. This gives us ${\cos(40^\circ - 10^\circ) = \cos(40^\circ)\cos(10^\circ) + \sin(40^\circ)\sin(10^\circ)}$. Suddenly, our complex expression is transformed into a much simpler form. This is the beauty of using trigonometric identities; they allow us to simplify expressions and make them easier to work with. It's like taking a tangled mess of wires and neatly organizing them so you can see what's going on.

Simplifying the Expression: A Step-by-Step Transformation

Now that we've identified the correct formula and made the appropriate substitutions, the next step is to simplify the expression. We've established that ${\cos(40^\circ)\cos(10^\circ) + \sin(40^\circ)\sin(10^\circ) = \cos(40^\circ - 10^\circ)}$. The only thing left to do is to perform the subtraction within the cosine function. This is a straightforward arithmetic operation, but it's crucial to get it right. Attention to detail is paramount in mathematics, as even a small error can lead to an incorrect answer. It's like building a house; you need to make sure each brick is placed correctly to ensure the structure is sound.

Subtracting 10 degrees from 40 degrees gives us 30 degrees. Therefore, our expression simplifies to ${\cos(30^\circ)}$. This is a significant step forward, as we've reduced a complex expression involving multiple trigonometric functions into a single, well-known value. The ability to simplify expressions is a fundamental skill in mathematics, and it's something that comes with practice and a solid understanding of the underlying principles. It's like learning to drive a car; the more you practice, the more comfortable and confident you become.

Finding the Equivalent Expression: The Grand Finale

We've successfully simplified the expression to ${\cos(30^\circ)}$. But wait, we're not quite done yet! The original question asks us to find an equivalent expression from a set of options. This means we need to recognize that ${\cos(30^\circ)}$ is a standard trigonometric value that can be expressed in a simplified form. Understanding the values of trigonometric functions for common angles, such as 30, 45, and 60 degrees, is essential for solving many problems. These values are like the alphabet of trigonometry; they form the foundation upon which more complex concepts are built.

Recall that ${\cos(30^\circ) = \frac{\sqrt{3}}{2}}$. This is a fundamental value that you should aim to memorize or be able to quickly derive using the unit circle or special right triangles. Having these values at your fingertips will save you time and effort on exams and in other problem-solving situations. It's like knowing your multiplication tables; it makes arithmetic much faster and easier.

Now, let's revisit the original options. We were given:

• A. ${\sin(40^\circ - 10^\circ)}$
• B. ${\cos(40^\circ + 10^\circ)}$

We've already determined that the equivalent expression is ${\cos(40^\circ - 10^\circ) = \cos(30^\circ)}$. So, let's evaluate the given options:

• A. ${\sin(40^\circ - 10^\circ) = \sin(30^\circ) = \frac{1}{2}}$
• B. ${\cos(40^\circ + 10^\circ) = \cos(50^\circ)}$

Comparing these results with our simplified expression, ${\cos(30^\circ) = \frac{\sqrt{3}}{2}}$, we can clearly see that neither option A nor option B is equivalent.

Wait a minute! It seems we've made a slight detour in our explanation and focused on finding the value of the expression. However, the question asks for the equivalent expression, not the numerical value. Let's rewind and focus on matching the simplified form, ${\cos(40^\circ - 10^\circ)}$, with the given options.

Looking back at the options:

• A. ${\sin(40^\circ - 10^\circ)}$
• B. ${\cos(40^\circ + 10^\circ)}$

We can see that the correct answer is the expression that matches the form ${\cos(A - B)}$, which we derived using the cosine angle subtraction formula. Therefore, the equivalent expression is:

B. ${\cos(40^\circ - 10^\circ)}$

Key Takeaways: Mastering Trigonometric Identities

This problem has been a fantastic journey through the world of trigonometric identities. We've seen how recognizing patterns, applying the correct formulas, and simplifying expressions are crucial skills in solving trigonometric problems. Mastering these skills will not only help you ace your exams but also give you a deeper appreciation for the beauty and elegance of mathematics. It's like learning a new language; the more you practice, the more fluent you become.

Here are some key takeaways to remember:

1. Recognize Trigonometric Identities: The cosine angle addition and subtraction formulas are essential tools in your trigonometric arsenal. Learn to recognize when these identities can be applied to simplify expressions.
2. Apply the Correct Formula: Choosing the right formula is crucial. In this case, the cosine angle subtraction formula was the key to unlocking the solution.
3. Simplify Expressions: Break down complex expressions into simpler forms by using trigonometric identities and algebraic manipulations.
4. Understand Standard Trigonometric Values: Knowing the values of trigonometric functions for common angles (30, 45, 60 degrees) will save you time and effort.
5. Pay Attention to the Question: Always make sure you're answering the question that's being asked. In this case, we needed to find the equivalent expression, not the numerical value.

Practice Makes Perfect: Your Path to Trigonometric Mastery

Like any skill, mastering trigonometry requires practice. The more problems you solve, the more comfortable you'll become with recognizing patterns, applying formulas, and simplifying expressions. Don't be afraid to make mistakes; they're a natural part of the learning process. The important thing is to learn from your mistakes and keep practicing.

So, go forth and conquer trigonometric expressions! With a solid understanding of the fundamentals and plenty of practice, you'll be able to tackle even the most challenging problems with confidence. Keep exploring, keep learning, and keep enjoying the fascinating world of mathematics! You've got this!

This article explores the trigonometric expression cos(40°)cos(10°) + sin(40°)sin(10°) and identifies its equivalent form using trigonometric identities. It provides a detailed explanation of the cosine angle subtraction formula and its application in simplifying the given expression. This guide aims to help students and enthusiasts understand trigonometric identities and improve their problem-solving skills in mathematics.