Solving Trigonometric Equations: Acute Angle θ in an Exemplary Case

Understanding how to solve trigonometric equations, particularly those involving acute angles, is a fundamental skill in mathematics. One specific example that challenges this understanding is when the equation [frac{cos theta - sin theta}{cos theta sin theta} frac{1 - sqrt{3}}{1 sqrt{3}}] is given for the acute angle θ. This article will guide you through the process step-by-step to find and understand the value of θ.

Step-by-Step Solution

To solve the equation, we first need to simplify the right-hand side of the expression. We do this by rationalizing the denominator using the conjugate of the denominator.

Rationalizing the Right Side

By multiplying both the numerator and the denominator by the conjugate of the denominator, we get:

[frac{1 - sqrt{3}}{1 sqrt{3}} cdot frac{1 - sqrt{3}}{1 - sqrt{3}} frac{(1 - sqrt{3})^2}{(1 sqrt{3})(1 - sqrt{3})}]

Simplifying the denominator:

[(1 sqrt{3})(1 - sqrt{3}) 1 - 3 -2]

Simplifying the numerator:

[(1 - sqrt{3})^2 1 - 2 sqrt{3} 3 4 - 2 sqrt{3}]

Thus, we have:

[frac{1 - sqrt{3}}{1 sqrt{3}} frac{4 - 2 sqrt{3}}{-2} -2 sqrt{3}]

So, our equation simplifies to:

[frac{cos theta - sin theta}{cos theta sin theta} -2 sqrt{3}]

Cross-Multiplying and Simplifying

Next, we cross-multiply to eliminate the fraction:

[cos theta - sin theta (-2 sqrt{3}) (cos theta sin theta)]

Expanding the right side:

[cos theta - sin theta -2 cos theta - 2 sin theta sqrt{3} cos theta sqrt{3} sin theta]

Rearranging gives:

[cos theta 2 sqrt{3} cos theta sin theta - sqrt{3} sin theta]

Factoring out the terms:

[(1 2 sqrt{3}) cos theta (1 - sqrt{3}) sin theta]

Dividing both sides by the common term:

[frac{1 2 sqrt{3}}{1 - sqrt{3}} tan theta]

To simplify the ratio, we multiply the numerator and denominator by the conjugate of the denominator:

[frac{1 2 sqrt{3}(1 - sqrt{3})(1 sqrt{3})}{1 - sqrt{3}(1 sqrt{3})} frac{1 2 sqrt{3} - sqrt{3} - 6}{1 - 3} frac{-5 2 sqrt{3}}{-2} sqrt{3}]

Thus, we have:

[tan theta sqrt{3}]

Finding the Angle

The angle θ for which tan θ √3 is:

[theta 60^{circ} text{ or } frac{pi}{3} text{ radians}]

Since θ is an acute angle, the solution is:

[theta 60^{circ}]

Conclusion

In this article, we have outlined a comprehensive step-by-step approach to solving the trigonometric equation involving the acute angle θ. The process involved rationalizing, cross-multiplying, and simplifying expressions. We have shown that the solution to the equation is θ 60°, reinforcing the importance of understanding trigonometric identities and ratios in solving complex equations.

Keywords

Trigonometric Equations, Acute Angle, Cosine-Sine Equation