For \(\cos 225^\circ\), since \(225^\circ = 180^\circ + 45^\circ\), we use the cosine angle addition identity:
["# Understanding (\cos 225^\circ) Using the Cosine Angle Addition Identity", "Cosine values for standard angles are foundational in trigonometry, but evaluating angles beyond the first quadrant—especially those greater than (90^\circ)—can be challenging. One powerful identity simplifies computing (\cos) for angles like (225^\circ): the cosine angle addition identity. In this article, we explore how to compute (\cos 225^\circ) using this identity, making it easier to understand the value and apply it across various trigonometric problems.", "## What is the Cosine Angle Addition Identity?", "The angle addition formula for cosine states:\n[\n\cos(a + b) = \cos a \cos b - \sin a \sin b\n]\nThis identity applies to any angles (a) and (b), enabling the expansion of (\cos) functions of sums into products of cosines and sines of the component angles. By rewriting (225^\circ) as (180^\circ + 45^\circ), we apply this identity to find the cosine value efficiently.", "## Applying the Identity to (\cos 225^\circ)", "Since (225^\circ = 180^\circ + 45^\circ), substitute (a = 180^\circ) and (b = 45^\circ) into the cosine addition formula:\n[\n\cos(180^\circ + 45^\circ) = \cos 180^\circ \cos 45^\circ - \sin 180^\circ \sin 45^\circ\n]", "Now evaluate each trigonometric function at these key angles:\n- (\cos 180^\circ = -1) (on the unit circle, 180° lies on the negative x-axis)\n- (\cos 45^\circ = \frac{\sqrt{2}}{2})\n- (\sin 180^\circ = 0) (sine is zero at 180°)\n- (\sin 45^\circ = \frac{\sqrt{2}}{2})", "Substitute these values:\n[\n\cos 225^\circ = (-1)\left(\frac{\sqrt{2}}{2}\right) - (0)\left(\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{2} - 0 = -\frac{\sqrt{2}}{2}\n]", "## Why This Identity Matters", "Using the cosine addition identity transforms a non-standard angle into a familiar expression involving well-known values. This method avoids memorizing every special cosine value and leverages symmetry and periodicity in the unit circle. Understanding this approach builds a foundation for computing other angles like (270^\circ), (315^\circ), or even angles beyond the first revolution.", "## Final Answer", "[\n\cos 225^\circ = -\frac{\sqrt{2}}{2}\n]", "---", "### Key Takeaways\n- Rewriting (225^\circ) as (180^\circ + 45^\circ) enables application of the cosine addition identity.\n- Known values of ( \cos 45^\circ = \frac{\sqrt{2}}{2} ), (\sin 45^\circ = \frac{\sqrt{2}}{2}), and ( \cos 180^\circ = -1 ), ( \sin 180^\circ = 0 ) simplify the evaluation.\n- This method applies broadly to angles expressed as sums involving standard angles.", "Mastering such identities enhances mathematical fluency and confidence when solving trigonometric problems involving larger or non-standard angles. Whether in academics, engineering, or physics, understanding how to decompose angles using additonal identities is a valuable skill that simplifies complex computations.", "---", "Keywords: (\cos 225^\circ), cosine addition formula, trigonometric identities, angle addition identity, unit circle, trigonometric values, mathematical computation, periodic functions."]
