Solving For Tan(2x-4) Given Sin(2x+15) = Cos(3x-10) A Trigonometric Approach

This article delves into the fascinating world of trigonometry, specifically focusing on solving trigonometric equations. We'll tackle a problem where we're given the equation sin(2x+15) = cos(3x-10) and our goal is to find the value of tan(2x-4) without resorting to calculators or mathematical tables. This exercise highlights the beauty of trigonometric identities and their power in simplifying complex problems.

Understanding the Core Concepts

Before diving into the solution, it's crucial to understand the fundamental trigonometric concepts that underpin our approach. The core idea revolves around the relationship between sine and cosine, particularly the cofunction identity. This identity states that sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). This relationship is derived from the complementary angles in a right-angled triangle. When two angles are complementary, their sum is 90 degrees. The sine of an angle is equal to the cosine of its complement, and vice-versa. Understanding this relationship is the key to unlocking the solution to our problem.

Furthermore, we need to be familiar with the properties of the tangent function. Tangent, denoted as tan(θ), is defined as the ratio of sine to cosine: tan(θ) = sin(θ) / cos(θ). The tangent function has a period of 180 degrees, meaning that tan(θ) = tan(θ + 180°n) where n is an integer. This periodicity is essential when finding general solutions to trigonometric equations. We should also be aware of specific values of tangent for common angles, such as tan(0°) = 0, tan(45°) = 1, and tan(30°) = 1/√3. While we are aiming to solve this problem without direct calculation, familiarity with these values can provide a useful context.

Finally, mastering the art of algebraic manipulation is critical in solving trigonometric equations. We'll need to rearrange equations, combine like terms, and apply trigonometric identities strategically. The goal is to isolate the variable x and eventually determine the value of tan(2x-4). The following steps will illustrate how these concepts are applied to solve the specific problem.

Applying the Cofunction Identity

Our starting point is the equation sin(2x + 15) = cos(3x - 10). To effectively use the cofunction identity, we can rewrite the cosine term as a sine term. Applying the identity cos(θ) = sin(90° - θ) to the right side of the equation, we get:

sin(2x + 15) = sin(90 - (3x - 10))

Simplifying the expression inside the sine function on the right side, we have:

sin(2x + 15) = sin(90 - 3x + 10)

sin(2x + 15) = sin(100 - 3x)

Now that we have sine functions on both sides of the equation, we can proceed to equate the angles. However, it's essential to remember that sine is periodic, meaning sin(θ) = sin(θ + 360°n) and sin(θ) = sin(180° - θ + 360°n) for any integer n. This periodicity leads to multiple possible solutions for x. We'll first consider the simpler case where the angles are directly equal:

2x + 15 = 100 - 3x

Solving for x

Now we have a simple linear equation in terms of x. To solve for x, we need to isolate it on one side of the equation. Adding 3x to both sides gives:

5x + 15 = 100

Subtracting 15 from both sides yields:

5x = 85

Finally, dividing both sides by 5, we find:

x = 17

This is one potential solution for x. However, we must also consider the second possibility arising from the periodic nature of sine, where sin(θ) = sin(180° - θ).

Considering the Second Case

Taking into account the identity sin(θ) = sin(180° - θ), we can set up another equation:

2x + 15 = 180 - (100 - 3x)

Simplifying the right side, we get:

2x + 15 = 180 - 100 + 3x

2x + 15 = 80 + 3x

Subtracting 2x from both sides gives:

15 = 80 + x

Subtracting 80 from both sides yields:

x = -65

So, we have two possible values for x: 17 and -65. However, to ensure we have the correct solution, we need to substitute these values back into the original equation or consider the context of the problem. In many trigonometric problems, the range of possible solutions is often restricted, which can help us eliminate extraneous solutions. For now, let's proceed with both values and see how they impact the final result.

Finding tan(2x-4)

Our ultimate goal is to find the value of tan(2x - 4). We now have two potential values for x, so we'll calculate tan(2x - 4) for each case.

Case 1: x = 17

Substituting x = 17 into the expression 2x - 4, we get:

2(17) - 4 = 34 - 4 = 30

Therefore, we need to find tan(30°). Recall that tan(θ) = sin(θ) / cos(θ). We know that sin(30°) = 1/2 and cos(30°) = √3/2. Thus,

tan(30°) = (1/2) / (√3/2) = 1/√3

Rationalizing the denominator, we multiply the numerator and denominator by √3:

tan(30°) = (1/√3) * (√3/√3) = √3/3

So, when x = 17, tan(2x - 4) = √3/3.

Case 2: x = -65

Substituting x = -65 into the expression 2x - 4, we get:

2(-65) - 4 = -130 - 4 = -134

Therefore, we need to find tan(-134°). Using the property that tan(θ) has a period of 180°, we can add 180° to -134° until we get an angle within the range of 0° to 180°.

-134° + 180° = 46°

So, tan(-134°) = tan(46°). While we don't know the exact value of tan(46°) without a calculator, this highlights the importance of understanding trigonometric identities and the periodicity of trigonometric functions.

Conclusion

Given the equation sin(2x + 15) = cos(3x - 10), we found two possible solutions for x: 17 and -65. When x = 17, tan(2x - 4) = tan(30°) = √3/3. When x = -65, tan(2x - 4) = tan(-134°) = tan(46°). This problem demonstrates the power of trigonometric identities, particularly the cofunction identity, in simplifying and solving trigonometric equations. Furthermore, it highlights the importance of considering the periodic nature of trigonometric functions to find all possible solutions. While we were able to find an exact value for tan(2x - 4) in one case, the other case required us to understand the properties of the tangent function and its periodicity. This comprehensive approach ensures a thorough understanding of the problem and its solution.

This exercise is a testament to the elegance and interconnectedness of mathematical concepts. By mastering these concepts, we can confidently tackle even the most challenging trigonometric problems. The ability to solve such problems without relying on calculators or tables underscores the deep understanding and appreciation of mathematical principles. This skill is not only valuable in academic settings but also in various real-world applications where a strong foundation in mathematics is essential.