Calculate Trigonometric Expression E = 2 Sin 30° + Tan 45° (sin 53° + Sin 37°)
In the realm of mathematics, trigonometric expressions often present a fascinating challenge. These expressions, which involve trigonometric functions like sine, cosine, and tangent, require a deep understanding of trigonometric identities and angle relationships. In this comprehensive guide, we will delve into the intricacies of evaluating a specific trigonometric expression, providing a step-by-step approach that will empower you to tackle similar problems with confidence. Our main focus will be on unraveling the expression:
E = 2 sen 30° + tg 45° (sen 53° + sen 37°)
This expression beautifully combines various trigonometric functions and angles, offering a rich learning experience. By dissecting each component and applying the appropriate trigonometric principles, we will systematically arrive at the final solution. So, buckle up and prepare for an enlightening journey into the world of trigonometric calculations!
Understanding the Fundamentals of Trigonometry
Before we embark on the calculation itself, it's crucial to lay a solid foundation by revisiting some fundamental concepts of trigonometry. Trigonometry, at its core, deals with the relationships between the sides and angles of triangles, particularly right-angled triangles. The three primary trigonometric functions – sine (sen), cosine (cos), and tangent (tg) – play a pivotal role in these relationships.
- Sine (sen): In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, we express this as sen θ = Opposite / Hypotenuse.
- Cosine (cos): The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In mathematical terms, cos θ = Adjacent / Hypotenuse.
- Tangent (tg): The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. We can write this as tg θ = Opposite / Adjacent.
These trigonometric functions have specific values for certain standard angles, such as 0°, 30°, 45°, 60°, and 90°. These values are essential to remember, as they frequently appear in trigonometric problems. For instance:
- sen 30° = 1/2
- cos 30° = √3/2
- tg 45° = 1
- sen 45° = √2/2
- cos 45° = √2/2
Moreover, understanding complementary angles is crucial for simplifying trigonometric expressions. Complementary angles are two angles that add up to 90°. A fundamental trigonometric identity states that the sine of an angle is equal to the cosine of its complement, and vice versa. Mathematically:
- sen θ = cos (90° - θ)
- cos θ = sen (90° - θ)
This identity will prove invaluable in simplifying the expression we are about to evaluate.
Step-by-Step Evaluation of the Expression
Now that we have refreshed our understanding of trigonometric fundamentals, let's embark on the journey of evaluating the expression:
E = 2 sen 30° + tg 45° (sen 53° + sen 37°)
We will tackle this expression step by step, breaking it down into manageable components.
Step 1: Evaluating 2 sen 30°
We know that sen 30° = 1/2. Therefore,
2 sen 30° = 2 * (1/2) = 1
So, the first part of the expression simplifies to 1.
Step 2: Evaluating tg 45°
As we discussed earlier, tg 45° = 1. This value is a cornerstone of trigonometric calculations and is worth memorizing.
Step 3: Evaluating (sen 53° + sen 37°)
This part of the expression requires a clever application of trigonometric identities. Notice that 53° and 37° are complementary angles, as 53° + 37° = 90°. We can use the complementary angle identity to simplify this expression.
Let's rewrite sen 53° using the identity sen θ = cos (90° - θ):
sen 53° = cos (90° - 53°) = cos 37°
Now, our expression becomes:
sen 53° + sen 37° = cos 37° + sen 37°
While we don't have exact values for sen 37° and cos 37° readily available, we can approximate them using the properties of a 3-4-5 right-angled triangle. In such a triangle, if one of the acute angles is approximately 37°, then the sides are in the ratio 3:4:5. Therefore:
sen 37° ≈ 3/5 = 0.6
cos 37° ≈ 4/5 = 0.8
So,
sen 53° + sen 37° ≈ 0.8 + 0.6 = 1.4
Step 4: Combining the Results
Now that we have evaluated each component of the expression, we can combine them to find the final answer:
E = 2 sen 30° + tg 45° (sen 53° + sen 37°)
E = 1 + 1 * (1.4)
E = 1 + 1.4
E = 2.4
Therefore, the value of the expression E is approximately 2.4.
Alternative Approach Using Trigonometric Identities
While we used approximations for sen 37° and cos 37° in the previous method, we can also solve this problem more precisely using the sum-to-product trigonometric identity.
The sum-to-product identity for the sum of sines states:
sen A + sen B = 2 sen ((A + B) / 2) cos ((A - B) / 2)
Applying this identity to sen 53° + sen 37°:
sen 53° + sen 37° = 2 sen ((53° + 37°) / 2) cos ((53° - 37°) / 2)
sen 53° + sen 37° = 2 sen (45°) cos (8°)
We know that sen 45° = √2/2. To find cos 8°, we would typically need a calculator or trigonometric tables. However, for the purpose of this example, let's assume we have the value of cos 8° ≈ 0.99.
Therefore:
sen 53° + sen 37° = 2 * (√2/2) * 0.99
sen 53° + sen 37° ≈ √2 * 0.99
sen 53° + sen 37° ≈ 1.4
As you can see, this method, even with the approximation of cos 8°, yields a similar result to our previous approximation. Now, let's combine the results as before:
E = 2 sen 30° + tg 45° (sen 53° + sen 37°)
E = 1 + 1 * (1.4)
E = 1 + 1.4
E = 2.4
Conclusion
In this comprehensive guide, we have successfully evaluated the trigonometric expression:
E = 2 sen 30° + tg 45° (sen 53° + sen 37°)
We explored two approaches: one involving approximations using the properties of a 3-4-5 triangle and another utilizing the sum-to-product trigonometric identity. Both methods led us to the same approximate answer of 2.4.
This exercise highlights the importance of understanding trigonometric fundamentals, including the definitions of trigonometric functions, standard angle values, and trigonometric identities. By mastering these concepts, you can confidently tackle a wide range of trigonometric problems.
Remember, practice is key to success in mathematics. So, keep exploring, keep practicing, and keep expanding your knowledge of trigonometry! This in-depth exploration of trigonometric calculations not only provides a solution to the given expression but also equips you with a versatile toolkit for approaching similar mathematical challenges. By understanding the underlying principles and practicing different solution strategies, you can enhance your problem-solving skills and gain a deeper appreciation for the elegance and power of mathematics.
