Solving the Equation 2sin(4x) - 3cos(4x) 0 within the Range [0, 2π]

Let's explore the solution to the equation 2sin(4x) - 3cos(4x) 0 within the range [0, 2π]. This problem involves trigonometric functions and their periodic properties. By using appropriate trigonometric identities and understanding the periodic nature of the tangent function, we can find all the values of x within the specified range.

Step-by-Step Solution

The given equation is:

2sin(4x) - 3cos(4x) 0

First, we can rewrite this equation as:

2sin(4x) 3cos(4x)

This can be further simplified as:

sin(4x) / cos(4x) 3 / 2

which can be expressed as:

tan(4x) 3 / 2

Taking the Inverse Tangent

To solve for 4x, we take the inverse tangent on both sides:

4x arctan(3 / 2) nπ

Here, n is an integer that accounts for the periodicity of the tangent function. The tangent function has a period of π, so it repeats its values every π units.

Expressing the Solution in Terms of x

Now, we can solve for x by dividing both sides by 4:

x (arctan(3 / 2) nπ) / 4

Thus, the general solution to the equation is:

x (1/4)(arctan(3 / 2) nπ)

Applying the Range [0, 2π]

Since we are interested in the values of x within the range [0, 2π], we need to determine the integer values of n that will give us solutions within this range. Let's find the upper limit for n.

2π (1/4)(arctan(3 / 2) nπ)

Multiplying both sides by 4:

8π arctan(3 / 2) nπ

Solving for n:

nπ 8π - arctan(3 / 2)

Therefore:

n 8 - (arctan(3 / 2) / π)

Since arctan(3 / 2) is approximately 0.9828 radians, we have:

Since n must be an integer, the possible values for n are from 0 to 7.

Final Values of x

Hence, the values of x that satisfy the equation within the range [0, 2π] are:

x (1/4)(arctan(3 / 2) nπ) where 0 ≤ n ≤ 7

Conclusion

This problem demonstrates the importance of understanding periodic functions and trigonometric identities. By using the tangent function and its properties, we can solve complex trigonometric equations step by step. The periodic nature of tangent, expressed by arctan, helps us find all possible solutions within any given range.

Related Keywords

trigonometric equations, solving trigonometric equations, periodic functions