Solving Trigonometric Expression: Sin35cos55 + 2cos55sin35 + 2cos60

Hey guys! Let's dive into this cool trigonometry problem. We're gonna break down the expression sin35cos55 + 2cos55sin35 + 2cos60. Don't worry, it looks a bit intimidating at first, but we'll tackle it step by step. We'll use some handy trig identities and a little bit of number crunching to get to the answer. Trigonometry might seem like something you only use in math class, but understanding it is super helpful for all sorts of things, from engineering to even just understanding how waves work. So, let's get started and make this expression simpler. The main goal here is to simplify the trigonometric expression and find its value. This involves using trigonometric identities to rewrite the expression in a more manageable form. Specifically, we'll aim to combine terms and use the known values of trigonometric functions for specific angles to arrive at the final solution. This process helps solidify your understanding of how these identities work and how they can be used to solve different kinds of mathematical problems. Ready to flex those math muscles? Let's go!

Breaking Down the Expression: The First Steps

Alright, let's look at the expression again: sin35cos55 + 2cos55sin35 + 2cos60. The first thing you might notice is that we have terms with sin and cos functions, and different angles (35 degrees, 55 degrees, and 60 degrees). Our strategy will be to use trigonometric identities to rewrite parts of the expression. This is where those identities you learned in class come in handy, like the angle sum and difference formulas, and the values of sine and cosine for special angles like 60 degrees. Let's start by looking at 2cos60. We know that cos60 is equal to 1/2. So, 2cos60 simplifies to 2 * (1/2) = 1. That's a nice easy start, right? Now, the expression becomes sin35cos55 + 2cos55sin35 + 1. See? Already a little cleaner. Next, we can rearrange the terms involving the sine and cosine functions. Using the commutative property of multiplication, 2cos55sin35 is the same as 2sin35cos55. This helps us group the terms in a way that might lead to something familiar. Keep your eye out for patterns! When you're dealing with trig problems, often the key is recognizing how different parts of the expression relate to each other through the identities. It's like a puzzle, and you're trying to fit all the pieces together. The more practice you get, the easier it becomes to spot those patterns. It's all about practice, guys!

Utilizing Trigonometric Identities: Angle Relationships

Now, let's look at the relationship between the angles. Notice that 35 degrees and 55 degrees are complementary angles (they add up to 90 degrees). Remember the cofunction identities? They tell us that sin(θ) = cos(90 - θ) and cos(θ) = sin(90 - θ). We can use these identities to rewrite either sin35 or cos55. For instance, since 55 = 90 - 35, we can say that cos55 = sin(90 - 55) = sin35. This is super helpful because it allows us to combine terms. Our expression now has sin35 appearing multiple times. Let's make that substitution: our expression sin35cos55 + 2cos55sin35 + 1 can be changed to sin35sin35 + 2sin35sin35 + 1. Or, in a more simplified form, we have sin^2(35) + 2sin^2(35) + 1. This looks much friendlier now! Do you see how using the identities helped us simplify things? It's like we're using shortcuts to solve the problem. Now, if we combine the sin^2 terms, we get 3sin^2(35) + 1. We're getting closer to a final answer.

Simplifying Further: Combining and Calculating

Okay, now we have the expression 3sin^2(35) + 1. Can we simplify this further? Well, we know the value of sin35, but it's not a common angle like 30, 45, or 60 degrees. So, we'll need to use a calculator to find the value of sin35. Using a calculator, sin35 ≈ 0.5736. Then, sin^2(35) ≈ (0.5736)^2 ≈ 0.3293. Substituting this back into the expression: 3 * 0.3293 + 1. Now, multiply 3 by 0.3293, which gives us approximately 0.9879. Finally, add 1 to that result. The final answer is approximately 1.9879. Keep in mind that this is an approximate value because we had to use an approximate value for sin35. However, we've successfully simplified the expression and found its approximate value. It's awesome how we broke down a complex-looking problem into simpler parts, using trig identities and basic arithmetic. This kind of problem-solving is fundamental in many areas of math and science, and it's all about practice and recognizing the patterns. Remember, the key is to stay organized, use the right formulas, and take it one step at a time. Awesome job, everyone!

Detailed Step-by-Step Solution

Let's go through the solution again, but this time with a more detailed step-by-step approach. This will help you solidify your understanding of each step and why we do it. Ready? Here we go!

1. Original Expression: sin35cos55 + 2cos55sin35 + 2cos60
2. Simplify 2cos60: Since cos60 = 1/2, 2cos60 = 1. Now we have: sin35cos55 + 2cos55sin35 + 1
3. Recognize Complementary Angles: 35 and 55 are complementary, so we can use cofunction identities. We know that cos55 = sin(90 - 55) = sin35.
4. Substitute cos55 with sin35: The expression now looks like this: sin35 * sin35 + 2 * sin35 * sin35 + 1, or sin^2(35) + 2sin^2(35) + 1.
5. Combine the sin^2 terms: This simplifies to 3sin^2(35) + 1.
6. Find the value of sin35: Using a calculator, we find that sin35 ≈ 0.5736.
7. Calculate sin^2(35): sin^2(35) ≈ (0.5736)^2 ≈ 0.3293.
8. Substitute and Calculate: 3 * 0.3293 + 1 ≈ 0.9879 + 1 ≈ 1.9879.
9. Final Answer: Approximately 1.9879.

The Importance of Trigonometric Identities

In this problem, the trigonometric identities were our secret weapons. They helped us rewrite the expression in a way that made it easier to solve. Trigonometric identities are equations that are true for all values of the variables involved. They are essential tools for simplifying expressions, solving equations, and proving other trigonometric relationships. Understanding and being able to apply these identities is crucial for anyone studying trigonometry. Here are a few key identities that are often used:

• Pythagorean Identities: sin^2(θ) + cos^2(θ) = 1
• Quotient Identities: tan(θ) = sin(θ)/cos(θ), cot(θ) = cos(θ)/sin(θ)
• Reciprocal Identities: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ)
• Angle Sum and Difference Formulas: These are used to find the sine, cosine, or tangent of sums or differences of angles. For instance, sin(A + B) = sinAcosB + cosAsinB
• Cofunction Identities: These relate trigonometric functions of complementary angles (angles that add up to 90 degrees), like we used in our problem: sin(θ) = cos(90 - θ) and cos(θ) = sin(90 - θ)

Knowing how to use these identities allows you to manipulate trigonometric expressions, simplify them, and solve complex problems more easily. They are the building blocks of many advanced concepts in trigonometry, calculus, and other areas of mathematics and physics. The more you practice using these identities, the more comfortable and confident you will become. It's like learning a new language - the more you use it, the easier it gets!

Practical Applications of Trigonometry

Okay, so we've solved a math problem, but where does trigonometry actually come into play in the real world? Well, it's used in a bunch of different fields. In navigation, trigonometry helps determine distances and directions, which is essential for things like GPS and mapping. Engineers use it to design bridges, buildings, and other structures, calculating angles and forces. In physics, trigonometry is fundamental for understanding wave motion, optics, and mechanics. Even in computer graphics and video game development, trigonometry helps create realistic visuals and animations. Imagine how the characters move and the world around them is created. It's all thanks to math, including trig! The core idea is that trigonometry provides the tools to relate angles and sides of triangles, which is vital in a wide range of applications. Think about the way a surveyor uses trigonometry to measure land, or how a pilot uses it to navigate an airplane. Trigonometry also helps determine the height of a building, or the distance to a star. So, even though it may seem abstract at times, trigonometry is a practical tool that helps us understand and interact with the world around us. So, the next time you see a bridge or play a video game, you can remember that trigonometry played a role in bringing it to life.

Tips for Solving Trigonometry Problems

Here are some tips to help you conquer future trigonometry problems:

• Memorize key identities: Knowing the basic identities is crucial. Make flashcards or create a cheat sheet to help you remember them.
• Practice, practice, practice: The more problems you solve, the more familiar you will become with different types of problems and how to approach them.
• Draw diagrams: Visualizing the problem with a diagram can help you understand the relationships between angles and sides.
• Identify the givens: Write down what you know (angles, side lengths) and what you need to find.
• Look for patterns: See if you can spot any special angles, complementary angles, or identities that can be used.
• Work systematically: Break down complex problems into smaller, manageable steps.
• Check your work: Make sure your answers make sense and double-check your calculations.

By following these tips, you'll be well on your way to mastering trigonometry. Remember, it's not always easy at first, but with practice and persistence, you'll become more confident in your ability to solve these problems. Keep going, and celebrate your progress along the way. You got this!

Conclusion: Wrapping Things Up

Great job, everyone! We successfully simplified and solved the trigonometric expression sin35cos55 + 2cos55sin35 + 2cos60. We used trigonometric identities, our knowledge of special angles, and a little bit of calculator work. We also explored some cool applications of trigonometry and got some helpful tips for future problems. Remember, practice is key, and don't be afraid to ask for help when you need it. Keep exploring the amazing world of math, and keep up the great work! You're all doing awesome!