Trigonometry Problem: Step-by-Step Solution For 11th Grade

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Trigonometry Problem: Step-by-Step Solution for 11th Grade

Hey guys! Let's break down this trigonometry problem step by step so everyone can follow along. We're tackling a question where the correct answer is E)3, and we need to understand exactly how to get there. No more head-scratching! Let's get started.

Understanding the Core Concepts

Before we dive into the nitty-gritty details of the problem, it's super important to make sure we're all on the same page with some fundamental trigonometry concepts. Knowing these concepts cold will make the problem way easier to handle.

  • Trigonometric Functions: At the heart of trigonometry are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Each relates an angle of a right triangle to the ratio of two of its sides.
  • Unit Circle: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It's an incredibly useful tool because the coordinates of any point on the unit circle are (cos θ, sin θ), where θ is the angle formed by the positive x-axis and the line connecting the origin to the point.
  • Trigonometric Identities: These are equations that are true for all values of the variables involved. Key identities include:
    • sin² θ + cos² θ = 1
    • tan θ = sin θ / cos θ
    • cot θ = cos θ / sin θ = 1 / tan θ
    • sec θ = 1 / cos θ
    • csc θ = 1 / sin θ
  • Angle Sum and Difference Formulas: These formulas help express trigonometric functions of sums and differences of angles:
    • sin (A + B) = sin A cos B + cos A sin B
    • sin (A - B) = sin A cos B - cos A sin B
    • cos (A + B) = cos A cos B - sin A sin B
    • cos (A - B) = cos A cos B + sin A sin B
  • Double-Angle Formulas: These are special cases of the sum formulas:
    • sin 2θ = 2 sin θ cos θ
    • cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ

Understanding these concepts thoroughly is essential. Trigonometry builds upon these foundations, so a solid grasp here makes tackling complex problems much more manageable. Think of these concepts as your trigonometry toolkit. Make sure you have the right tools before you start building!

Deconstructing the Problem

Okay, let's assume we have a specific 11th-grade trigonometry problem that leads to the answer E)3. Since the original question wasn't fully provided, I’ll create a sample problem that incorporates common 11th-grade trigonometry concepts and demonstrates a step-by-step solution. This way, you will understand how the answer E)3 can be reached. Make sure you follow each and every steps carefully.

Sample Problem:

Simplify the following expression:

(2sin(x)cos(x)) / (sin(x) + cos(x))^2 - 1

Given that x = π/4, find the value of the simplified expression.

Step 1: Simplify the Trigonometric Expression

Our first goal is to simplify the given expression. Let's start by recognizing that 2sin(x)cos(x) is the double-angle formula for sine, i.e., sin(2x). So, we can rewrite the expression as:

sin(2x) / ((sin(x) + cos(x))^2 - 1)

Step 2: Expand the Denominator

Next, we need to expand the denominator (sin(x) + cos(x))^2. Using the formula (a + b)^2 = a^2 + 2ab + b^2, we get:

(sin(x) + cos(x))^2 = sin^2(x) + 2sin(x)cos(x) + cos^2(x)

Step 3: Use the Pythagorean Identity

We know that sin^2(x) + cos^2(x) = 1. So, we can simplify the denominator further:

sin^2(x) + 2sin(x)cos(x) + cos^2(x) = 1 + 2sin(x)cos(x) = 1 + sin(2x)

Now, the original expression becomes:

sin(2x) / (1 + sin(2x) - 1)

Step 4: Simplify the Expression Further

We can now simplify the expression by canceling out the 1s in the denominator:

sin(2x) / sin(2x)

This simplifies to 1, provided that sin(2x) is not equal to zero.

Step 5: Substitute the Given Value of x

We are given that x = π/4. Let's substitute this value into our simplified expression:

2x = 2 * (π/4) = π/2

So we need to find sin(π/2).

Step 6: Evaluate sin(π/2)

We know that sin(π/2) = 1. Therefore, sin(2x) = 1 when x = π/4.

Step 7: Substitute Back into the Simplified Expression

Now we substitute sin(2x) = 1 back into our simplified expression sin(2x) / (sin(x) + cos(x))^2 - 1:

1 / (1 + 1 - 1) = 1 / 1 = 1

Step 8: Re-evaluate with a different problem to achieve a solution of 3.

Okay, our initial sample problem simplified to 1, not 3. To achieve the desired answer of 3, we need a different, more complex trigonometric expression. Let's try this one:

Simplify: (3cos(2x) + 6sin^2(x)) / cos(2x) when x = π/6

Step 9: Simplify the New Trigonometric Expression

We're starting with: (3cos(2x) + 6sin^2(x)) / cos(2x)

Notice that we can rewrite 6sin^2(x) as 3 * 2sin^2(x). Also, remember the identity cos(2x) = 1 - 2sin^2(x). Therefore, 2sin^2(x) = 1 - cos(2x). Substitute this into our expression:

(3cos(2x) + 3(1 - cos(2x))) / cos(2x)

Step 10: Distribute and Simplify

Distribute the 3 in the numerator:

(3cos(2x) + 3 - 3cos(2x)) / cos(2x)

The 3cos(2x) and -3cos(2x) terms cancel out:

3 / cos(2x)

Step 11: Substitute the Value of x

We are given that x = π/6. So, 2x = 2 * (π/6) = π/3.

We need to find cos(π/3).

Step 12: Evaluate cos(π/3)

We know that cos(π/3) = 1/2

Step 13: Substitute and Calculate the Final Result

Substitute cos(π/3) = 1/2 back into our simplified expression:

3 / (1/2) = 3 * 2 = 6

Let's try adjusting the original problem again to hit our target of E)3.

Modified Sample Problem:

Simplify: (3cos(2x)) / (2cos^2(x) - 1) when x = π/3

Step 1: Recognize the Identity

Notice that 2cos^2(x) - 1 is a common identity and is equal to cos(2x). So we have:

(3cos(2x)) / cos(2x)

Step 2: Simplify

We can cancel out the cos(2x) terms (assuming cos(2x) is not zero):

3

Step 3: Substitute x = π/3 (but it's not really necessary)

Since the expression simplifies directly to 3, the value of 'x' doesn't actually matter in this specific (contrived) example. Regardless, let's verify cos(2x) isn't zero.

2x = 2 * (π/3) = (2π)/3

cos((2π)/3) = -1/2 which is NOT zero.

Final Answer: 3

Key Takeaways

  • Simplify Before Substituting: Always try to simplify the trigonometric expression as much as possible before plugging in the value of x. This reduces the chances of making errors and makes the calculations easier.
  • Know Your Identities: A strong command of trigonometric identities is crucial. They are the tools that allow you to manipulate and simplify expressions.
  • Step-by-Step Approach: Break down the problem into smaller, manageable steps. This makes the problem less daunting and easier to follow.
  • Double-Check: Always double-check your work, especially when dealing with substitutions and simplifications.

By following these steps and understanding the underlying concepts, you'll be well-equipped to tackle even the trickiest 11th-grade trigonometry problems! Remember, practice makes perfect, so keep at it!