Solving Trigonometric Equation: tan x = cot(3x - 30°)

We are given the equation: $ \tan x = \cot(3x - 30^\circ) $ To solve for $x$, we can use the trigonometric identity $\cot \theta = \tan(90^\circ - \theta)$.

Applying Trigonometric Identity

Substitute the identity into the equation:

$ \tan x = \tan \left( 90^\circ - (3x - 30^\circ) \right) $

Simplify the angle:

$ \tan x = \tan \left( 90^\circ - 3x + 30^\circ \right) $

$ \tan x = \tan \left( 120^\circ - 3x \right) $

Finding the Value of x

If $\tan A = \tan B$, then the general solution is $A = B + n \cdot 180^\circ$, where $n$ is an integer.

Equating the angles:

$ x = (120^\circ - 3x) + n \cdot 180^\circ $

Now, solve for $x$:

• Combine terms with $x$: $x + 3x = 120^\circ + n \cdot 180^\circ$
• Simplify: $4x = 120^\circ + n \cdot 180^\circ$
• Divide by 4: $x = \frac{120^\circ}{4} + \frac{n \cdot 180^\circ}{4}$
• General solution: $x = 30^\circ + n \cdot 45^\circ$

Determining the Specific Solution

We need to find a value of $x$ that matches one of the given options. Let's test integer values for $n$:

• For $n = 0$: $x = 30^\circ + 0 \cdot 45^\circ = 30^\circ$
• For $n = 1$: $x = 30^\circ + 1 \cdot 45^\circ = 75^\circ$
• For $n = -1$: $x = 30^\circ - 1 \cdot 45^\circ = -15^\circ$

The value $x = 30^\circ$ is one of the options and satisfies the equation.

Verification

Check if $x = 30^\circ$ works:

• Left Hand Side (LHS): $\tan(30^\circ) = \frac{1}{\sqrt{3}}$
• Right Hand Side (RHS): $\cot(3(30^\circ) - 30^\circ) = \cot(90^\circ - 30^\circ) = \cot(60^\circ) = \frac{1}{\sqrt{3}}$

Since LHS = RHS, the solution $x = 30^\circ$ is correct.