Trigonometric Equations and Their Solutions: sin7x cos11x

Trigonometry is a vital branch of mathematics, often cropping up in a variety of real-world scenarios and problem-solving situations. One of the famous properties of trigonometric functions involves the relationships between sine, cosine, tangent, and cotangent. Today, we'll explore a specific problem: given sin7x cos11x, what is the value of tan9x cot9x? This problem will guide us through the application of trigonometric identities and the simplification of expressions.

Understanding the Given Equation: sin7x cos11x

The problem statement provides the equation sin7x cos11x. The key to solving this equation is understanding the relationship between sine and cosine functions. One of the fundamental trigonometric identities is that cosθ sin(90° - θ). This identity allows us to convert the given equation into a more familiar form:

sin7x cos11x

sin7x sin(90° - 11x)

Using Trigonometric Identities to Solve for x

Let's proceed with the given equation:

sin7x sin(90° - 11x)

From the above equation, we can deduce that the angles inside the sine functions must be equal or supplementary:

7x 90° - 11x or 7x 180° - (90° - 11x)

First, let's consider the simplest case (without supplementary angles):

7x 90° - 11x

Adding 11x to both sides...

18x 90°

Dividing both sides by 18...

x 5°

Finding the Value of tan9x cot9x

Now that we have found the value of x, let's use it to find the value of tan9x cot9x:

Since x 5°, we need to find the value of 9x:

9x 9 × 5° 45°

We now need to calculate tan45° and cot45°:

tan45° 1 (since tan45° 1 is a well-known trigonometric value)

cot45° 1 (since cot45° 1 / tan45° 1 / 1 1)

Thus, the value of tan9x cot9x becomes:

tan45° cot45° 1 × 1 2

Conclusion

To summarize, given the equation sin7x cos11x, we were able to determine that x 5°. Subsequently, we calculated the value of 9x, which is 45°, and found that tan9x cot9x equals 2. This example showcases the application of trigonometric identities and the simplification of trigonometric expressions to solve complex equations.

Keywords

trigonometry, trigonometric equations, tangent cotangent

By understanding the application of trigonometric identities and simplifying expressions, we can tackle a variety of problems in trigonometry. This guide has provided a detailed step-by-step solution to the given equation, demonstrating the process of solving such equations and the use of trigonometric identities. Further exploration in trigonometry can help deepen your understanding and readiness for more complex problems.