Solving Trigonometric Equations: sin 3x cos 3x 1/2

In this comprehensive guide, we will explore how to solve the trigonometric equation sin 3x cos 3x 1/2. This process involves the application of trigonometric identities and solving for the variable x. Let's break down the solution step-by-step.

Step-by-Step Solution

The given equation is:

Apply the Trigonometric Identity: The identity we will use is sin a cos a (1/√2) sin(a π/4). For a 3x, substituting in the identity, we get:

sin 3x cos 3x  (1/√2) sin(3x   π/4)

Simplify the Equation: Substituting the identity into the original equation, we have:

(1/√2) sin(3x   π/4)  1/2

Divide both sides by (1/√2):

sin(3x   π/4)  (1/2) * (1/√2)  (1/√2) * (√2/2)  1/2√2  √2/4

Solve for sin(3x π/4): To find the angles for which sin(3x π/4) √2/4, we need to use the general solution for sin y k: y arcsin(k) 2nπ or y π - arcsin(k) 2nπ, where n is an integer. Let:

3x   π/4  arcsin(√2/4)   2nπ or 3x   π/4  π - arcsin(√2/4)   2nπ

Solve for x: For the first case, solve for x:

3x  arcsin(√2/4) - π/4   2nπ

Dividing by 3:

x  (arcsin(√2/4) - π/4   2nπ)/3

For the second case, solve for x:

3x  π - arcsin(√2/4) - π/4   2nπ

Dividing by 3:

x  (π - arcsin(√2/4) - π/4   2nπ)/3

General Solutions for x

The general solutions for x are:

x (arcsin(√2/4) - π/4 2nπ)/3 x (π - arcsin(√2/4) - π/4 2nπ)/3 Substitute different integer values for n to find specific values of x.

To find a numerical solution, you can calculate arcsin(√2/4) using a calculator.

Alternative Method Using Another Identity

We can also use the identity cos x cos y 2 cos(x/2 y/2) cos(x/2 - y/2).

Note: Recognize that sin3x cos3x π/2. This gives us:

cos3x π/2 cos3x  -sqrt(2) cos(3x 3π/4)  1/2

Solve for cos(3x 3π/4): Isolating the cosine term, we get:

cos(3x 3π/4)  -1/(4√2)

Solve for 3x: The cosine function gives us:

3x 3π/4  ± arccos(-1/(4√2))   k?2π

Solving for x:

x  ± (1/3) arccos(-1/(4√2)) - π/4   k?(2/3)π

With k in Z, this provides a general solution for x. This alternative method also confirms the trigonometric solutions previously found.

Conclusion

In summary, solving the equation sin 3x cos 3x 1/2 involves the application of trigonometric identities and careful algebraic manipulation. By using the identities and solving for the variable x, we arrived at the general solutions for x. These solutions can be further explored by substituting different integer values for n or calculating specific numerical values with the help of a calculator.