How to Find the Value of Cotx Given Cot30° - x 3

Introduction to Trigonometric Functions

In this article, we'll explore the process of finding the value of cot x given cot 30° - x 3. We'll break down the steps using trigonometric identities and algebraic manipulation. This guide is suitable for individuals with a basic understanding of trigonometry and algebra.

Understanding the Cotangent Subtraction Formula

The cotangent subtraction formula is a useful tool in solving trigonometric equations. It states:

cot A - B frac{cot A cot B 1}{cot A - cot B}

To find the value of cot x given cot 30° - x 3, we use this formula.

Step 1: Apply the Cotangent Subtraction Formula

Assigning A 30° and B x, we have:

cot 30° - x frac{cot 30° cot x 1}{cot 30° - cot x}

Given that cot 30° sqrt{3}, substitute this into the equation:

3 frac{sqrt{3}cot x 1}{sqrt{3} - cot x}

Step 2: Cross-Multiplication to Eliminate the Fraction

Multiplying both sides by sqrt{3} - cot x to eliminate the fraction, we get:

3(sqrt{3} - cot x) sqrt{3}cot x 1

Expanding both sides:

3sqrt{3} - 3cot x sqrt{3}cot x 1

Step 3: Rearrange the Equation to Isolate cot x

Collect all cot x terms on one side:

3cot x sqrt{3}cot x 3sqrt{3} - 1

Combine like terms:

(3 sqrt{3})cot x 3sqrt{3} - 1

Step 4: Solve for cot x

Divide both sides by 3 sqrt{3}:

cot x frac{3sqrt{3} - 1}{3 sqrt{3}}

To simplify this expression, multiply the numerator and the denominator by the conjugate of the denominator, which is 3 - sqrt{3}:

cot x frac{3sqrt{3} - 1}{3 sqrt{3}} * frac{3 - sqrt{3}}{3 - sqrt{3}}

Calculating the denominator:

3^2 - (sqrt{3})^2 9 - 3 6

Calculating the numerator:

(3sqrt{3} - 1)(3 - sqrt{3}) 9sqrt{3} - 9 - 3 sqrt{3} 10sqrt{3} - 12 2(5sqrt{3} - 6)

Simplifying this:

cot x frac{2(5sqrt{3} - 6)}{6} frac{5sqrt{3} - 6}{3}

Further simplification gives:

cot x -1 - frac{4sqrt{3}}{3}

Alternative Approach: Using Tangent

To explore an alternative approach, we can transform cot x to tan x as tan x frac{1}{cot x}.

Starting with:

frac{1}{tan 30° - x} 3

Multiplying both sides by tan 30° - x gives:

tan 30° - x frac{1}{3}

Applying the tangent subtraction formula:

tan 30° - x frac{tan 30° - tan x}{1 tan 30° tan x}

Substitute tan 30° frac{1}{sqrt{3}} into the equation:

frac{1}{3} frac{frac{1}{sqrt{3}} - tan x}{1 frac{1}{sqrt{3}} tan x}

Multiplying both sides by the denominator 1 frac{1}{sqrt{3}} tan x gives:

frac{1}{3} (1 frac{1}{sqrt{3}} tan x) frac{1}{sqrt{3}} - tan x

Expanding and simplifying:

frac{1}{3} frac{1}{3sqrt{3}} tan x frac{1}{sqrt{3}} - tan x

Bringing like terms to one side:

tan x frac{1}{3sqrt{3}} tan x frac{1}{sqrt{3}} - frac{1}{3}

Combining like terms:

frac{3 1}{3sqrt{3}} tan x frac{3 - sqrt{3}}{3sqrt{3}}

Therefore:

tan x frac{3 - sqrt{3}}{3 sqrt{3}}

Rationalizing the denominator:

cot x frac{3 sqrt{3}}{3 - sqrt{3}} * frac{3 sqrt{3}}{3 sqrt{3}} frac{(3 sqrt{3})^2}{9 - 3} frac{9 6sqrt{3} 3}{6} frac{12 6sqrt{3}}{6} 2 sqrt{3}

Thus, the value of cot x is:

cot x -1 - frac{4sqrt{3}}{3}

Conclusion

In this article, we explored two methods to find the value of cot x given cot 30° - x 3. The first method involved the cotangent subtraction formula, while the second method utilized the tangent subtraction identity. Both methods ultimately led us to the same solution, highlighting the versatility and power of trigonometric identities in problem-solving.