Solving the Trigonometric Equation: 2cos(x)tan(x)sin(x) = 0
This article will guide you through the complete solution for the trigonometric equation 2cos(x)tan(x)sin(x) = 0. We'll break down the process step-by-step, ensuring you understand each stage. Mastering this type of problem is crucial for anyone studying trigonometry, precalculus, or calculus.
Understanding the Equation
The equation 2cos(x)tan(x)sin(x) = 0 involves three trigonometric functions: cosine (cos(x)), tangent (tan(x)), and sine (sin(x)). To solve it, we'll leverage the zero product property: if the product of several factors is zero, then at least one of the factors must be zero.
Applying the Zero Product Property
Our equation can be rewritten as:
2 * cos(x) * tan(x) * sin(x) = 0
According to the zero product property, this equation is true if at least one of the following is true:
- cos(x) = 0
- tan(x) = 0
- sin(x) = 0
We'll solve each of these simpler equations individually.
Solving cos(x) = 0
The cosine function, cos(x), equals zero at x = Ο/2 + nΟ, where 'n' is any integer. This means:
- x = Ο/2 + nΟ (where n is any integer)
Solving tan(x) = 0
The tangent function, tan(x), equals zero at x = nΟ, where 'n' is any integer. Therefore:
- x = nΟ (where n is any integer)
Solving sin(x) = 0
The sine function, sin(x), is equal to zero at x = nΟ, where 'n' is any integer. This gives us:
- x = nΟ (where n is any integer)
Combining the Solutions
Notice that the solutions for tan(x) = 0 and sin(x) = 0 are the same: x = nΟ. Therefore, the complete solution to the original equation, 2cos(x)tan(x)sin(x) = 0, is the union of the solutions we found:
- x = nΟ (where n is any integer)
- x = Ο/2 + nΟ (where n is any integer)
This means the equation is satisfied for all values of x that are multiples of Ο and for all values of x that are odd multiples of Ο/2.
Verification and Further Exploration
You can verify these solutions by substituting them back into the original equation. Try plugging in a few values of 'n' to confirm they satisfy the equation.
This problem provides a great foundation for understanding trigonometric equations and solving them systematically. You can extend this understanding by tackling more complex trigonometric equations involving multiple angles or identities. Remember, practice is key to mastering these concepts.
