Solving Trigonometric Equations: A Step-by-Step Guide

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Hey guys! Let's dive into the fascinating world of trigonometry and break down how to solve equations involving trigonometric functions. In this article, we're going to tackle the equation tg30 × ctg60 + tg60 × ctg30 = 10/3. This might look intimidating at first, but don't worry, we'll take it one step at a time and make sure you understand every part of the process. So, grab your calculators and let's get started!

Understanding the Basics: Tangent (tg) and Cotangent (ctg)

Before we jump into solving the equation, let's quickly refresh our understanding of the trigonometric functions involved: tangent (tg) and cotangent (ctg). These functions are fundamental to trigonometry, and knowing their definitions and properties is crucial for solving trigonometric equations.

  • Tangent (tg): The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, we write it as tg(θ) = Opposite / Adjacent. Think of it as the "rise over run" if you're familiar with slope.

  • Cotangent (ctg): The cotangent of an angle is the reciprocal of the tangent. It's the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle. So, ctg(θ) = Adjacent / Opposite, which is also equal to 1 / tg(θ). Basically, it's the inverse tangent.

Key Angles and Their Values

To solve our equation, we need to know the values of tg and ctg for some key angles, specifically 30° and 60°. These angles appear frequently in trigonometric problems, so it's a good idea to memorize their values. You can derive these values using the special right triangles (30-60-90 triangles) or from the unit circle.

  • tg(30°): The tangent of 30 degrees is 1/√3, which can also be written as √3/3 after rationalizing the denominator. Remember this! It’s a common value.

  • ctg(60°): The cotangent of 60 degrees is the reciprocal of the tangent of 60 degrees. Since tg(60°) = √3, then ctg(60°) = 1/√3 = √3/3. Notice that ctg(60°) has the same value as tg(30°).

  • tg(60°): The tangent of 60 degrees is √3. This is another key value to keep in mind.

  • ctg(30°): The cotangent of 30 degrees is the reciprocal of the tangent of 30 degrees. Since tg(30°) = 1/√3, then ctg(30°) = √3. Just like tg(60°), ctg(30°) is equal to √3.

Knowing these values is like having the right tools in your toolbox – they'll help you solve the equation much more efficiently. Let's move on to the next step!

Breaking Down the Equation: tg30 × ctg60 + tg60 × ctg30 = 10/3

Now that we've got our trigonometric values sorted, let's break down the given equation: tg30 × ctg60 + tg60 × ctg30 = 10/3. Our goal here is to verify if this equation holds true, meaning we want to check if the left-hand side (LHS) actually equals the right-hand side (RHS), which is 10/3.

Substituting the Values

The first thing we need to do is substitute the values we know for tg30, ctg60, tg60, and ctg30 into the equation. We've already established that:

  • tg(30°) = √3/3
  • ctg(60°) = √3/3
  • tg(60°) = √3
  • ctg(30°) = √3

So, let's plug these values into the LHS of our equation:

LHS = (√3/3) × (√3/3) + √3 × √3

See how we're just replacing the trigonometric functions with their numerical values? This is a crucial step in simplifying the equation. Next, we'll perform the multiplication.

Performing the Multiplication

Now, let's perform the multiplication in the expression. We have two terms to multiply:

  • (√3/3) × (√3/3)
  • √3 × √3

Let's tackle the first one. When you multiply fractions, you multiply the numerators (the top numbers) and the denominators (the bottom numbers). So,

(√3/3) × (√3/3) = (√3 × √3) / (3 × 3)

Remember that √3 × √3 = 3, so we get:

(√3 × √3) / (3 × 3) = 3 / 9 = 1/3

Now, let's move on to the second term:

√3 × √3 = 3

It's that simple! Now we have the results of both multiplications, and we can move on to the next step: addition.

Simplifying the Expression: Addition and Verification

Okay, we've done the multiplication, and now we need to add the results together. Remember, our LHS is now:

LHS = 1/3 + 3

Adding the Terms

To add these terms, we need to have a common denominator. We can rewrite 3 as a fraction with a denominator of 3:

3 = 9/3

Now we can easily add the fractions:

LHS = 1/3 + 9/3 = (1 + 9) / 3 = 10/3

Verification

Guess what? We've arrived at a very important moment! We've simplified the left-hand side (LHS) of our equation to 10/3. Now, let's compare this to the right-hand side (RHS) of the original equation:

Original Equation: tg30 × ctg60 + tg60 × ctg30 = 10/3

Our simplified LHS: 10/3

RHS: 10/3

LHS = RHS

The equation holds true!

We have successfully shown that the left-hand side of the equation is indeed equal to the right-hand side. This means our initial equation is valid.

Conclusion: Mastering Trigonometric Equations

Great job, guys! We've successfully tackled the equation tg30 × ctg60 + tg60 × ctg30 = 10/3. By breaking it down step-by-step, understanding the basic trigonometric functions, and substituting the correct values, we were able to verify that the equation holds true. This is a testament to the power of a systematic approach in solving mathematical problems.

Remember, the key to mastering trigonometric equations is practice. The more problems you solve, the more comfortable you'll become with the concepts and the techniques involved. So, keep practicing, and don't be afraid to tackle challenging problems. You've got this!

Key Takeaways

  • Understand the definitions of trigonometric functions: Tangent (tg) and cotangent (ctg) are ratios of sides in a right-angled triangle.
  • Memorize key values: Knowing the values of tg and ctg for common angles like 30°, 60°, and 45° is extremely helpful.
  • Break down the equation: Simplify complex equations by substituting values and performing operations step-by-step.
  • Verify your results: Always check if the left-hand side equals the right-hand side to ensure your solution is correct.

Keep exploring the world of trigonometry, and you'll discover even more fascinating concepts and applications. Happy solving!