\cos heta = rac{-12}{(2\sqrt{6})(2\sqrt{6})} = rac{-12}{24} = -rac{1}{2} \Rightarrow heta = rac{2\pi}{3}

["# Understanding ( \cos \ heta = -\frac{1}{2} ): Finding the Angle ( \ heta ) in Radians", "Solving trigonometric equations is a fundamental skill in mathematics, especially for students and enthusiasts exploring angles, waves, and periodic phenomena. One common problem involves determining the angle ( \ heta ) where ( \cos \ heta = -\frac{1}{2} ). In this article, we’ll explore the derivation, simplify the expression, and determine the exact value of ( \ heta ), highlighting key concepts of trigonometric values and radian measures.", "## The Equation: ( \cos \ heta = -\frac{12}{(2\sqrt{6})(2\sqrt{6})} = -\frac{1}{2} )", "We begin with the cosine equation:\n[\n\cos \ heta = -\frac{12}{(2\sqrt{6})(2\sqrt{6})}\n]\nFirst, simplify the denominator:\n[\n(2\sqrt{6})(2\sqrt{6}) = 4 \cdot (\sqrt{6})^2 = 4 \cdot 6 = 24\n]\nSo the expression simplifies cleanly to:\n[\n\cos \ heta = -\frac{12}{24} = -\frac{1}{2}\n]", "This confirms the trigonometric equation:\n[\n\cos \ heta = -\frac{1}{2}\n]", "## Why ( \cos \ heta = -\frac{1}{2} )? Understanding Cosine Values", "The cosine of an angle corresponds to the x-coordinate on the unit circle. Values of ( \cos \ heta = -\frac{1}{2} ) occur at specific standard angles, reflecting symmetry in the unit circle. These occur particularly at angles where the reference angle is ( \frac{\pi}{3} ) radians (60°), but in quadrants where cosine is negative — the second and third quadrants.", "Specifically:\n- ( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} )\n- ( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} )", "So, the general solutions for ( \ heta ) are angles coterminal with ( \frac{2\pi}{3} ) and ( \frac{4\pi}{3} ) in the interval ( [0, 2\pi) ).", "## Determining the Principal Value: ( \ heta = \frac{2\pi}{3} )", "Among these, the smallest positive solution — the principal value commonly accepted in standard trigonometric references — is:\n[\n\ heta = \frac{2\pi}{3}\n]", "This angle lies in the second quadrant, where cosine is negative, and its reference angle is ( \frac{\pi}{3} ). The value ( \frac{2\pi}{3} ) radians equals ( 120^\circ ), a well-known angle in geometry.", "## Confirming the Solution: Plugging Back into Cosine", "To verify, compute:\n[\n\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\n]\nThis matches our simplified equation, confirming the solution is correct.", "## Why This Angle Matters: Geometry and Applications", "The angle ( \ heta = \frac{2\pi}{3} ) appears frequently in equilateral triangles, rotations, and physical wave interference. For instance:\n- In a 30-60-90 triangle, angles like ( \frac{2\pi}{3} ) (120°) help explain vector directions and force components.\n- In complex numbers, ( e^{i\frac{2\pi}{3}} ) corresponds to rotation by 120°, crucial in signal processing and engineering.", "## Conclusion: The Exact Value and Its Significance", "From the equation ( \cos \ heta = -\frac{1}{2} ), we derived that the simplest principal solution is:\n[\n\ heta = \frac{2\pi}{3}\n]\nThis result reinforces key trigonometric principles — using algebra, simplifying radicals, and interpreting unit circle behavior. Understanding such identities helps in solving complex equations and modeling periodic patterns in science and engineering.", "Whether you’re studying for exams, building foundational math skills, or applying trigonometry in real-world contexts, mastering how to trace equations from cosine values back to angles is essential — and solving ( \cos \ heta = -\frac{1}{2} ) is a perfect starting point.", "Try practicing with other cosine values — each equation brings clarity to the beautiful symmetry of the trigonometric world!"]