Mastering Double-Angle Identities Rewriting 2cos75sin75

Hey everyone! Today, we're diving into a classic trigonometry problem that involves double-angle identities. These identities are super useful for simplifying trigonometric expressions and solving equations. Let's break down the problem and see how we can crack it.

The Problem: Rewriting $2 \cos 75^{\circ} \sin 75^{\circ}$

Our mission, should we choose to accept it, is to rewrite the expression $2 \cos 75^{\circ} \sin 75^{\circ}$ using a double-angle identity. We're given a few options, and we need to figure out which one is the correct transformation.

Before we jump into the solution, let's refresh our memory on double-angle identities. These are trigonometric identities that express trigonometric functions of double angles (like $2x$) in terms of trigonometric functions of the single angle ($x$).

Double-Angle Identities: A Quick Refresher

The most common double-angle identities are:

• Sine: $\sin(2x) = 2 \sin(x) \cos(x)$
• Cosine: There are three forms for cosine:
  • $\cos(2x) = \cos^2(x) - \sin^2(x)$
  • $\cos(2x) = 2 \cos^2(x) - 1$
  • $\cos(2x) = 1 - 2 \sin^2(x)$
• Tangent: $\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}$

For our problem, the sine double-angle identity looks particularly promising. Notice how it has the form $2 \sin(x) \cos(x)$, which is exactly what we have in our expression!

Cracking the Code: Applying the Double-Angle Identity

Now, let's apply the sine double-angle identity to our expression:

$2 \cos 75^{\circ} \sin 75^{\circ}$

Comparing this to the identity $\sin(2x) = 2 \sin(x) \cos(x)$, we can see that our $x$ is $75^{\circ}$. So, we can rewrite the expression as:

$\sin(2 \cdot 75^{\circ}) = \sin(150^{\circ})$

And there we have it! We've successfully rewritten the expression using the double-angle identity.

Why is Understanding Double-Angle Identities Important?

Double-angle identities, guys, are super important tools in trigonometry and calculus. They help simplify complex trigonometric expressions, which can be useful in a bunch of situations, like:

• Solving Trigonometric Equations: Double-angle identities allow us to rewrite equations in a form that's easier to solve.
• Integration: Some integrals involving trigonometric functions can be simplified using these identities.
• Simplifying Expressions: As we saw in our problem, they can make complicated expressions much simpler.
• Real-World Applications: Trigonometry, and thus these identities, pop up in fields like physics, engineering, and navigation.

A Deep Dive into Sine Double-Angle Identity

The sine double-angle identity, $\sin(2x) = 2 \sin(x) \cos(x)$, is a cornerstone of trigonometric identities and deserves a closer look. Understanding its derivation and applications can significantly enhance your problem-solving skills in trigonometry and related fields. Let’s delve deeper into this identity.

Derivation of the Sine Double-Angle Identity

The sine double-angle identity can be derived directly from the sine addition formula. The sine addition formula states that:

$\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)$

To derive the double-angle identity, we simply let $A = x$ and $B = x$. Substituting these values into the addition formula, we get:

$\sin(x + x) = \sin(x) \cos(x) + \cos(x) \sin(x)$

$\sin(2x) = 2 \sin(x) \cos(x)$

This straightforward derivation highlights the connection between fundamental trigonometric identities and showcases how one can be derived from another. This process of derivation not only reinforces the understanding of the identities but also provides a deeper appreciation for the interconnectedness of mathematical concepts.

Applications and Problem-Solving

The sine double-angle identity is a powerful tool for simplifying expressions and solving trigonometric equations. Its application can transform seemingly complex problems into manageable forms. Here are a few key areas where this identity shines:

1. Simplification of Trigonometric Expressions:

   When faced with expressions involving products of sines and cosines, the double-angle identity can be used to reduce the expression to a single trigonometric function. For instance, consider the expression $4 \sin(\theta) \cos(\theta)$. Recognizing the pattern $2 \sin(x) \cos(x)$ from the double-angle identity, we can rewrite the expression as $2 \cdot 2 \sin(\theta) \cos(\theta) = 2 \sin(2\theta)$. This simplification is crucial in various contexts, including calculus and physics.

2. Solving Trigonometric Equations:

   The identity plays a vital role in solving trigonometric equations where double angles are involved. By rewriting the equation using the double-angle identity, the equation can be transformed into a simpler form that is easier to solve. For example, consider the equation $\sin(2x) = \cos(x)$. Using the identity, we can rewrite this as $2 \sin(x) \cos(x) = \cos(x)$. This form allows us to solve for $x$ by factoring or using other algebraic techniques.

3. Integration in Calculus:

   In calculus, the sine double-angle identity is invaluable for integrating functions involving trigonometric expressions. Integrals that seem challenging at first can be simplified by applying the identity. For instance, integrals of the form $\int \sin(x) \cos(x) dx$ can be easily solved by substituting $u = \sin(x)$ or $u = \cos(x)$. However, the integral $\int 2 \sin(x) \cos(x) dx$ becomes even more straightforward when recognized as $\int \sin(2x) dx$, which can be directly integrated.

4. Complex Numbers and Euler's Formula:

   In the realm of complex numbers, the sine double-angle identity can be related to Euler's formula, which connects complex exponentials and trigonometric functions. Euler's formula states that $e^{ix} = \cos(x) + i \sin(x)$. By considering $e^{i2x}$, we can express it as $\cos(2x) + i \sin(2x)$. Using the identity, $\sin(2x)$ can be expanded, providing a deeper insight into the relationships between complex exponentials and trigonometric functions.

5. Applications in Physics and Engineering:

   The sine double-angle identity finds practical applications in physics and engineering, particularly in areas such as wave mechanics and signal processing. For instance, in wave mechanics, the identity can be used to analyze interference patterns and wave superposition. In signal processing, it helps in decomposing and synthesizing signals, which is fundamental in communications and audio processing.

6. Geometric Applications:

   The identity also has geometric applications, especially in problems involving triangles and circles. For example, in right triangles, the double-angle identity can be used to find trigonometric ratios of angles that are twice the size of other known angles. This is particularly useful in problems involving angle bisection and trigonometric proofs.

Common Mistakes to Avoid

When working with double-angle identities, it’s easy to slip up if you're not careful. Here are a few common mistakes to watch out for:

• Confusing the Identities: Make sure you're using the correct identity! It's easy to mix up the sine and cosine identities, so double-check before you apply them.
• Forgetting the Coefficient: The sine double-angle identity is $**\sin(2x) = 2 \sin(x) \cos(x)**$. Don't forget the '2'!
• Incorrect Substitution: When substituting, be sure you're replacing the entire expression correctly. It's a good idea to write out each step to avoid errors.
• Not Simplifying Fully: Once you've applied the identity, make sure you simplify the resulting expression as much as possible.

Practice Makes Perfect

The best way to master double-angle identities is to practice, practice, practice! Try working through different problems, and don't be afraid to make mistakes. Each mistake is a learning opportunity.

Example Problems

Let's try a couple more examples to solidify our understanding:

Example 1: Simplify $\sin(x) \cos(x)$.

We can rewrite this as $\frac{1}{2} \cdot 2 \sin(x) \cos(x)$. Now, using the double-angle identity, we get $\frac{1}{2} \sin(2x)$.

Example 2: If $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in the first quadrant, find $\sin(2\theta)$.

First, we need to find $\cos(\theta)$. Since $\theta$ is in the first quadrant, $\cos(\theta)$ is positive. Using the Pythagorean identity, $\sin^2(\theta) + \cos^2(\theta) = 1$, we get:

$(\frac{3}{5})^2 + \cos^2(\theta) = 1$

$\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$

$\cos(\theta) = \frac{4}{5}$

Now, we can use the double-angle identity:

$\sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$

Conclusion: Mastering Double-Angle Identities

Double-angle identities are powerful tools that can simplify trigonometric expressions and solve equations. By understanding the identities and practicing their application, you'll be well-equipped to tackle a wide range of trigonometric problems.

Remember, guys, the key is to recognize the patterns and apply the identities correctly. Keep practicing, and you'll become a double-angle identity pro in no time!

So, to recap, the correct answer to our original problem, rewriting $2 \cos 75^{\circ} \sin 75^{\circ}$, is C. $\sin 150^{\circ}$. Great job, everyone!