Find The Numerical Value Of The Expression E = (Tg² 30° - Sec² 45°) / (Cos 60° + Tg 45°). Possible Answers: A) -2/3, B) 2/3, C) 3/2.

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In this article, we will delve into the process of evaluating a complex trigonometric expression. Trigonometric expressions often involve various trigonometric functions such as tangent (Tg), secant (Sec), and cosine (Cos) applied to specific angles. These expressions can seem daunting at first, but with a systematic approach and a solid understanding of trigonometric values for common angles, they can be simplified and evaluated accurately. Our focus will be on dissecting the given expression, identifying the trigonometric values for the specified angles, and performing the necessary arithmetic operations to arrive at the final numerical value. By breaking down the problem into smaller, manageable steps, we will not only solve the specific expression but also gain a deeper appreciation for the elegance and interconnectedness of trigonometric functions. This exercise will enhance our problem-solving skills in trigonometry and provide a foundation for tackling more complex mathematical challenges. Let's embark on this journey of trigonometric evaluation and unravel the numerical value of the given expression.

Problem Statement

We are tasked with finding the numerical value of the expression:

E = (Tg² 30° - Sec² 45°) / (Cos 60° + Tg 45°)

This expression involves the tangent (Tg), secant (Sec), and cosine (Cos) functions evaluated at specific angles: 30°, 45°, and 60°. To solve this, we need to recall the values of these trigonometric functions for these common angles and then substitute them into the expression. The expression involves squaring, subtraction, addition, and division, so we must follow the order of operations (PEMDAS/BODMAS) to arrive at the correct answer. This problem serves as a good exercise in applying trigonometric identities and arithmetic operations to simplify and evaluate a mathematical expression. By carefully substituting the values and performing the calculations, we can determine the numerical value of E and gain a deeper understanding of trigonometric functions.

The possible answers are:

A) -2/3

B) 2/3

C) 3/2

Solution

Step 1: Recall Trigonometric Values

First, we need to remember the values of the trigonometric functions for the given angles:

  • Tg 30° = 1/√3
  • Sec 45° = √2
  • Cos 60° = 1/2
  • Tg 45° = 1

These values are fundamental in trigonometry and can be derived from the unit circle or special right triangles (30-60-90 and 45-45-90 triangles). It is crucial to memorize these values or have a quick method to derive them, as they frequently appear in trigonometric problems. The tangent of 30 degrees is the ratio of the opposite side to the adjacent side in a 30-60-90 triangle, resulting in 1/√3. The secant of 45 degrees, which is the reciprocal of cosine, is √2. The cosine of 60 degrees is 1/2, and the tangent of 45 degrees is 1, representing the equal sides in a 45-45-90 triangle. Having these values at our fingertips will enable us to efficiently solve the given expression.

Step 2: Substitute the Values

Now, substitute these values into the expression:

E = ((1/√3)² - (√2)²) / (1/2 + 1)

This step involves replacing the trigonometric functions with their corresponding numerical values. It is essential to be meticulous in this step to avoid any errors. We replace Tg 30° with 1/√3, Sec 45° with √2, Cos 60° with 1/2, and Tg 45° with 1. The expression now consists of purely numerical values and arithmetic operations. This substitution allows us to move from a trigonometric expression to an algebraic one, which can be simplified using standard arithmetic rules. The next step will involve simplifying the squares and performing the addition in the denominator, bringing us closer to the final numerical value of the expression. This process highlights the importance of accurate substitution in mathematical problem-solving.

Step 3: Simplify the Expression

Simplify the squares:

E = (1/3 - 2) / (1/2 + 1)

Here, we square the values in the numerator. (1/√3)² becomes 1/3, and (√2)² becomes 2. This simplification is a crucial step in reducing the complexity of the expression. Squaring the values removes the square roots and makes the numbers easier to work with. The expression now involves simple fractions and integers, making it more amenable to further simplification. The next step will involve finding a common denominator for the fractions in the numerator and denominator, which will allow us to perform the addition and subtraction operations. This process demonstrates the importance of basic arithmetic skills in simplifying mathematical expressions and ultimately finding their numerical values.

Step 4: Find Common Denominators and Combine Fractions

Find a common denominator in the numerator and denominator:

E = ((1 - 6)/3) / ((1 + 2)/2)

In this step, we find a common denominator to combine the fractions in both the numerator and the denominator. In the numerator, the common denominator is 3, so we rewrite 2 as 6/3. In the denominator, the common denominator is 2, so we rewrite 1 as 2/2. This allows us to combine the fractions: (1/3 - 2) becomes (1/3 - 6/3) = (1 - 6)/3, and (1/2 + 1) becomes (1/2 + 2/2) = (1 + 2)/2. Finding a common denominator is a fundamental skill in fraction arithmetic, enabling us to add or subtract fractions correctly. This step simplifies the expression further, making it easier to perform the division in the next step. The ability to manipulate fractions is essential in various mathematical contexts, including trigonometry, algebra, and calculus.

Step 5: Continue Simplifying

Continue simplifying:

E = (-5/3) / (3/2)

Now, we perform the subtraction and addition in the numerator and denominator, respectively. (1 - 6) equals -5, so the numerator becomes -5/3. (1 + 2) equals 3, so the denominator becomes 3/2. This step consolidates the fractions into single values, making the division operation more straightforward. The expression is now in the form of a fraction divided by another fraction. To perform this division, we will multiply the numerator by the reciprocal of the denominator, a standard procedure in fraction arithmetic. This step brings us closer to the final numerical value of the expression, demonstrating the step-by-step approach to simplifying complex mathematical expressions.

Step 6: Divide the Fractions

To divide fractions, multiply by the reciprocal:

E = (-5/3) * (2/3)

Dividing by a fraction is equivalent to multiplying by its reciprocal. So, we change the division operation to multiplication and invert the second fraction (3/2 becomes 2/3). This is a fundamental rule of fraction arithmetic. The expression now involves the multiplication of two fractions. To multiply fractions, we multiply the numerators together and the denominators together. This step transforms the division problem into a multiplication problem, which is often easier to handle. The next step will involve performing the multiplication and simplifying the resulting fraction, leading us to the final numerical value of the expression.

Step 7: Multiply the Fractions

Multiply the fractions:

E = -10/9

We multiply the numerators (-5 * 2 = -10) and the denominators (3 * 3 = 9). This gives us the fraction -10/9. This fraction represents the numerical value of the expression E. In this case, the fraction is already in its simplest form, as 10 and 9 have no common factors other than 1. Therefore, -10/9 is the final simplified result. This step completes the evaluation of the trigonometric expression, demonstrating the process of substituting trigonometric values, simplifying the expression using arithmetic operations, and arriving at the final numerical answer. The result highlights the importance of accuracy and attention to detail in mathematical problem-solving.

Step 8: Final Answer

However, the answer -10/9 is not among the options provided. Let's re-evaluate our calculations to find any potential errors.

Going back to Step 3:

E = (1/3 - 2) / (1/2 + 1)

E = ((1 - 6)/3) / ((1 + 2)/2)

E = (-5/3) / (3/2)

E = (-5/3) * (2/3)

E = -10/9

It seems there was an error in the provided options. The correct answer is -10/9.

Let's reassess the given options. It appears there might be a discrepancy between the calculated result (-10/9) and the provided choices. A thorough review of each step is essential to ensure accuracy. Starting from the beginning:

  • Step 1: Trigonometric values are correctly identified: Tg 30° = 1/√3, Sec 45° = √2, Cos 60° = 1/2, Tg 45° = 1.
  • Step 2: Substitution is accurate: E = ((1/√3)² - (√2)²) / (1/2 + 1).
  • Step 3: Squaring is correct: E = (1/3 - 2) / (1/2 + 1).
  • Step 4: Common denominators are found correctly: E = ((1 - 6)/3) / ((1 + 2)/2).
  • Step 5: Simplification is accurate: E = (-5/3) / (3/2).
  • Step 6: Division by reciprocal is correctly applied: E = (-5/3) * (2/3).
  • Step 7: Multiplication yields E = -10/9.

Upon meticulous re-evaluation, each step is confirmed to be accurate. Therefore, the calculated result of -10/9 is indeed correct. The discrepancy lies within the provided options, indicating a possible error in the answer choices. In such cases, it is crucial to trust the calculated result and recognize that the error may exist in the given options. This highlights the importance of both accurate calculation and critical evaluation of results in mathematical problem-solving.

Final Answer:

The correct answer is not listed in the options. The numerical value of the expression E is -10/9.