Calculating Trigonometric Expressions A Step-by-Step Guide

In mathematics, trigonometric expressions often appear in various contexts, from geometry problems to physics applications. Evaluating these expressions requires a solid understanding of trigonometric functions and their values for specific angles. In this article, we will delve into the process of calculating a given trigonometric expression step by step, reinforcing key concepts and techniques along the way. Specifically, we will address the expression:

$E=2 \operatorname{sen} 30^{\circ}+\operatorname{tg} 45^{\circ}(\operatorname{sen} 53^{\circ}+\operatorname{sen} 37^{\circ})$

This expression involves sine ($\operatorname{sen}$) and tangent ($\operatorname{tg}$) functions evaluated at different angles. Our goal is to simplify this expression to arrive at a numerical answer, choosing from the provided options: a) 1.2, b) 3.2, c) 3.4, d) 4.5, and e) 52. Let's embark on this calculation journey, breaking down each component and applying relevant trigonometric identities and values.

Understanding Trigonometric Functions and Values

Before diving into the calculation, it's crucial to understand the fundamental trigonometric functions: sine, cosine, and tangent. These functions relate the angles of a right triangle to the ratios of its sides. The sine of an angle is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. Moreover, memorizing the values of these functions for some standard angles like 0°, 30°, 45°, 60°, and 90° is extremely beneficial for solving trigonometric problems. These values can be derived from the unit circle or special right triangles (30-60-90 and 45-45-90 triangles).

For our problem, we need to recall the values of $\operatorname{sen} 30^{\circ}$ and $\operatorname{tg} 45^{\circ}$. From the unit circle or special right triangles, we know that $\operatorname{sen} 30^{\circ} = \frac{1}{2}$ and $\operatorname{tg} 45^{\circ} = 1$. We also need to consider the values of $\operatorname{sen} 53^{\circ}$ and $\operatorname{sen} 37^{\circ}$. These angles are not standard angles, but they are related in a useful way. Notice that 53° and 37° are complementary angles, meaning their sum is 90°. This relationship allows us to use the cofunction identity, which states that the sine of an angle is equal to the cosine of its complement. In other words, $\operatorname{sen} x = \cos(90^{\circ} - x)$. Therefore, $\operatorname{sen} 53^{\circ} = \cos 37^{\circ}$ and $\operatorname{sen} 37^{\circ} = \cos 53^{\circ}$. Furthermore, angles 37° and 53° often appear in 3-4-5 right triangles, where the sides are in the ratio 3:4:5. In such a triangle, $\operatorname{sen} 37^{\circ} = \frac{3}{5}$ and $\cos 37^{\circ} = \frac{4}{5}$, which means $\operatorname{sen} 53^{\circ} = \frac{4}{5}$.

With these values in mind, we are well-equipped to tackle the calculation of the expression. The ability to recall and apply these trigonometric values efficiently is a cornerstone of solving more complex problems in trigonometry and related fields. Understanding the relationships between angles and their trigonometric functions not only simplifies calculations but also deepens our understanding of the mathematical principles at play.

Step-by-Step Calculation of the Expression

Now, let's proceed with the step-by-step calculation of the expression:

$E=2 \operatorname{sen} 30^{\circ}+\operatorname{tg} 45^{\circ}(\operatorname{sen} 53^{\circ}+\operatorname{sen} 37^{\circ})$

Step 1: Substitute known trigonometric values

We know that $\operatorname{sen} 30^{\circ} = \frac{1}{2}$, $\operatorname{tg} 45^{\circ} = 1$, $\operatorname{sen} 53^{\circ} \approx \frac{4}{5}$, and $\operatorname{sen} 37^{\circ} \approx \frac{3}{5}$. Substituting these values into the expression, we get:

$E = 2 \cdot \frac{1}{2} + 1 \cdot (\frac{4}{5} + \frac{3}{5})$

This substitution simplifies the expression by replacing the trigonometric functions with their numerical equivalents. It's a crucial step in transforming the expression into a more manageable form, paving the way for arithmetic operations.

Step 2: Simplify the expression

Next, we perform the arithmetic operations:

$E = 1 + (\frac{4}{5} + \frac{3}{5})$

$E = 1 + \frac{7}{5}$

Here, we first multiplied 2 by $\frac{1}{2}$ to get 1. Then, we added the fractions $\frac{4}{5}$ and $\frac{3}{5}$, which have a common denominator, resulting in $\frac{7}{5}$. This step demonstrates the importance of following the order of operations (PEMDAS/BODMAS) to ensure accurate calculations.

Step 3: Convert to a decimal or mixed number

To compare the result with the given options, we convert the fraction to a decimal:

$E = 1 + 1.4$

$E = 2.4$

Alternatively, we can express the result as a mixed number:

$E = 1 + \frac{7}{5} = \frac{5}{5} + \frac{7}{5} = \frac{12}{5} = 2\frac{2}{5}$

Both the decimal and mixed number representations provide the same value, allowing us to clearly see the final result. This step is vital for matching our calculated answer with the available choices.

Step 4: Compare with the given options

Now, we compare our calculated value (2.4) with the given options:

a) 1.2 b) 3.2 c) 3.4 d) 4.5 e) 52

None of the given options match our calculated value of 2.4 exactly. However, it appears there might be a mistake in the original problem or the given options. Let's re-evaluate our steps to ensure we haven't made any errors.

Re-evaluating the Calculation and Identifying Potential Errors

Upon reviewing our calculation, we find no arithmetic errors in the steps we took. The trigonometric values used were also correct. However, the absence of a matching option suggests a possible issue with the problem statement or the answer choices provided.

Let's consider a slight modification to the values to see if it leads to one of the provided options. Suppose the expression was slightly different, or the values of $\operatorname{sen} 53^{\circ}$ and $\operatorname{sen} 37^{\circ}$ were approximated differently. If we use the approximation $\operatorname{sen} 53^{\circ} \approx 0.8$ and $\operatorname{sen} 37^{\circ} \approx 0.6$, which are common approximations, the calculation would proceed as follows:

$E = 2 \cdot \frac{1}{2} + 1 \cdot (0.8 + 0.6)$

$E = 1 + 1.4$

$E = 2.4$

The result remains 2.4, which still doesn't match any of the options. This reinforces the likelihood of an error in the original question or options.

Another approach is to consider the possibility of a typo in the options themselves. If option b) was intended to be 2.4 instead of 3.2, then it would match our calculated answer. However, without further information or clarification, we cannot definitively say this is the case.

In situations like this, it's essential to communicate the discrepancy to the source of the problem, whether it's a teacher, textbook, or exam. Pointing out the inconsistency allows for correction and ensures that students are not penalized for a problem with an incorrect answer set.

Conclusion and Key Takeaways

In this article, we meticulously calculated the trigonometric expression:

$E=2 \operatorname{sen} 30^{\circ}+\operatorname{tg} 45^{\circ}(\operatorname{sen} 53^{\circ}+\operatorname{sen} 37^{\circ})$

Our step-by-step calculation, using the known trigonometric values and approximations, resulted in a value of 2.4. However, this value did not match any of the provided options (a) 1.2, b) 3.2, c) 3.4, d) 4.5, and e) 52). This discrepancy highlights the importance of verifying results and recognizing potential errors in problem statements or answer choices.

Key takeaways from this exercise include:

1. Understanding Trigonometric Values: A strong foundation in the values of trigonometric functions for standard angles (0°, 30°, 45°, 60°, 90°) is crucial for solving trigonometric expressions efficiently. Additionally, knowing approximations for angles like 37° and 53° can be helpful.
2. Applying Trigonometric Identities: The cofunction identity ($\operatorname{sen} x = \cos(90^{\circ} - x)$) and other trigonometric identities are valuable tools for simplifying expressions and relating different trigonometric functions.
3. Following the Order of Operations: Adhering to the order of operations (PEMDAS/BODMAS) is essential for accurate calculations. Performing operations in the correct sequence prevents errors and ensures the correct result.
4. Verifying Results: Always double-check your calculations and compare your answer with the given options. If a discrepancy arises, re-evaluate your steps and consider the possibility of errors in the problem statement or answer choices.
5. Communicating Discrepancies: If you encounter a problem with no matching answer, it's important to communicate this to the appropriate source. This allows for correction and ensures fairness in assessments.

Trigonometric calculations are a fundamental aspect of mathematics and have applications in various fields. By mastering the concepts and techniques discussed in this article, you will be well-equipped to tackle a wide range of trigonometric problems. Remember to practice regularly and develop a keen eye for detail to minimize errors and maximize accuracy. The systematic approach to problem-solving, as demonstrated in this article, will not only help you in mathematics but also in other areas of study and in real-world situations.