Calculate E = Tan(30°) * Sec(60°) - Sin(37°) * Cos(30°) / Sin²(45°) Step-by-Step Guide

Hey guys! Let's break down this math problem step by step. We're tasked with calculating the value of 'E' given the trigonometric expression:

E = Tan(30°) * Sec(60°) - Sin(37°) * Cos(30°) / Sin²(45°)

This looks a bit intimidating at first, but don't worry! We'll tackle it by finding the values of each trigonometric function individually and then plugging them back into the equation. Remember, understanding the basics of trigonometry is key here. We'll be using some common trigonometric values that you might have memorized or can easily find in a trig table. Let's dive in!

Breaking Down the Expression

First, let's identify the trigonometric functions we need to evaluate. We have tangent (Tan), secant (Sec), sine (Sin), and cosine (Cos) at various angles. Knowing the definitions of these functions is crucial. Tangent is the ratio of sine to cosine, secant is the reciprocal of cosine, sine and cosine are fundamental trigonometric ratios, and we'll also encounter sine squared. Breaking down the expression into smaller, manageable parts makes the entire problem less overwhelming. We'll take each term one by one, find its value, and then combine them according to the original equation. This approach not only simplifies the calculation but also helps in understanding the logic behind each step. So, let's start with the first term, Tan(30°).

1. Evaluating Tan(30°)

Okay, so what is the tangent of 30 degrees? Remember that Tan(θ) = Sin(θ) / Cos(θ). We know that Sin(30°) = 1/2 and Cos(30°) = √3/2. Therefore:

Tan(30°) = (1/2) / (√3/2) = 1/√3

To rationalize the denominator, we multiply both the numerator and denominator by √3:

Tan(30°) = (1/√3) * (√3/√3) = √3/3

So, we've got our first value! The tangent of 30 degrees is √3/3. This is a common trigonometric value, and it's super helpful to have it memorized. But if you don't, no worries – you can always derive it using the sine and cosine values. This step highlights the importance of understanding the relationship between different trigonometric functions. Now, let's move on to the next term in our expression: Sec(60°).

2. Figuring Out Sec(60°)

Next up, we need to find the secant of 60 degrees. Remember, secant is the reciprocal of cosine: Sec(θ) = 1/Cos(θ). We know that Cos(60°) = 1/2. So:

Sec(60°) = 1 / (1/2) = 2

Easy peasy! The secant of 60 degrees is 2. This one's relatively straightforward once you remember the relationship between secant and cosine. It's like a quick win in our calculation journey. Now, we're making good progress. We've tackled Tan(30°) and Sec(60°). Let's move on to the next part of the expression, which involves sine and cosine of different angles. This is where things might get a little more interesting, but we've got this!

3. Calculating Sin(37°)

Alright, let's tackle Sin(37°). Now, 37 degrees isn't one of our standard angles (like 30°, 45°, or 60°), but it's a common angle that often appears in trigonometry problems. Sin(37°) is approximately 0.6. You might have this memorized, or you might need to use a calculator or a trigonometric table to find this value. It's a good idea to be familiar with this approximation, as it can save you time on exams or when solving problems. This value is particularly useful in physics and engineering applications as well. So, keep this one in your mental toolkit! Now, let's move on to Cos(30°).

4. Finding Cos(30°)

We've already encountered Cos(30°) earlier when we were calculating Tan(30°), but let's reiterate it here. Cos(30°) = √3/2. This is another one of those fundamental trigonometric values that's super useful to know. If you've worked with special right triangles (30-60-90 triangles), this value should be pretty familiar to you. It's like an old friend in the world of trigonometry! Now that we have this value, we're one step closer to solving our main equation. Let's move on to the final trigonometric function we need to evaluate: Sin²(45°).

5. Determining Sin²(45°)

Last but not least, we need to find Sin²(45°). First, let's find Sin(45°). Sin(45°) = √2/2. This is another common trigonometric value that comes from the special right triangle (45-45-90 triangle). Now, we need to square this value:

Sin²(45°) = (√2/2)² = 2/4 = 1/2

So, Sin²(45°) = 1/2. We've now evaluated all the individual trigonometric functions in our expression. Give yourself a pat on the back! This was a significant step. Now comes the exciting part: plugging these values back into the original equation and simplifying. This is where all our hard work pays off!

Plugging Values Back into the Equation

Okay, we've got all our individual trigonometric values. Let's plug them back into the original equation:

E = Tan(30°) * Sec(60°) - Sin(37°) * Cos(30°) / Sin²(45°)

Substituting the values we found:

E = (√3/3) * 2 - 0.6 * (√3/2) / (1/2)

Now, let's simplify this expression step by step. Remember, order of operations (PEMDAS/BODMAS) is our best friend here. First, we'll handle the multiplication and division. Then, we'll take care of the subtraction. Let's start with the first term: (√3/3) * 2.

6. Simplifying the Expression

Let's simplify the first part of the equation: (√3/3) * 2

(√3/3) * 2 = 2√3/3

Now, let's move on to the second part of the equation: 0.6 * (√3/2) / (1/2). This part involves multiplication and division, so we'll tackle it carefully.

0.  6 * (√3/2) / (1/2) = 0.6 * (√3/2) * 2 = 0.6√3

Now we have:

E = 2√3/3 - 0.6√3

To combine these terms, we need a common denominator. Let's convert 0.6 to a fraction: 0.6 = 6/10 = 3/5. So we have:

E = 2√3/3 - (3/5)√3

Now, let's find a common denominator for 3 and 5, which is 15. We'll rewrite the fractions:

E = (10√3/15) - (9√3/15)

Now we can subtract the terms:

E = (10√3 - 9√3) / 15 = √3/15

So, the final value of E is √3/15. We did it! This might seem like a long journey, but we broke the problem down into manageable steps and conquered each one. The value of E is √3/15. This entire process underscores the importance of breaking down complex problems into smaller, more manageable steps. By tackling each trigonometric function individually and then combining the results, we made the calculation much simpler. And that's the beauty of problem-solving in mathematics!

Final Answer

Therefore, E = √3/15. Woohoo! We've successfully calculated the value of E. This problem was a great exercise in applying our knowledge of trigonometric functions and algebraic manipulation. The final answer is E = √3/15. Remember, the key to solving these types of problems is to break them down into smaller steps, evaluate each part carefully, and then combine the results. You've got this!