Solution: $\tan 45^\circ = 1$. $\sin 315^\circ = \sin(360^\circ - 45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2}$. Therefore, the expression becomes:
["SEO Optimized Article: Understanding the Trigonometric Solution $\ an 45^\circ = 1$ and the Identity $\sin 315^\circ = -\frac{\sqrt{2}}{2}$", "---", "### Unlocking Trigonometric Truths: How $\ an 45^\circ = 1$ Powers Key Calculations", "Trigonometry forms the backbone of many mathematical, engineering, and physics applications. Among its fundamental truths, the identity $\ an 45^\circ = 1$ stands as a crisp and essential principle. But what happens when this principle is extended to solve more complex expressions like $\sin 315^\circ$? In this article, we explore how fundamental trigonometric identities connect to real-world mathematical applications—especially in evaluating angles beyond the first quadrant.", "---", "## Why $\ an 45^\circ = 1$ Matters in Trigonometry", "The value $\ an 45^\circ = 1$ arises directly from the unit circle and the properties of a 45-45-90 triangle. In such a triangle, the opposite and adjacent sides are equal, making the tangent (ratio of opposite to adjacent) exactly 1. This identity not only reinforces exact values but also supports key transformations in trigonometric simplifications and problem-solving.", "---", "## Solving $\sin 315^\circ$: A Step-by-Step Breakdown", "While $\ an 45^\circ = 1$ is straightforward, trigonometric values at different angles—particularly those outside the first quadrant—require careful application of identities. Consider $\sin 315^\circ$:", "### Step 1: Reference Angle\n$315^\circ$ lies in the fourth quadrant, where sine values are negative. To find the reference angle, subtract from $360^\circ$:\n[\n360^\circ - 315^\circ = 45^\circ\n]\nSo, $315^\circ = 360^\circ - 45^\circ$, making $45^\circ$ its reference angle.", "### Step 2: Apply the Negative Sine Identity\nSince $\sin(360^\circ - \ heta) = -\sin \ heta$, we substitute:\n[\n\sin 315^\circ = \sin(360^\circ - 45^\circ) = -\sin 45^\circ\n]", "### Step 3: Use the Exact Value of $\sin 45^\circ$\nFrom standard trigonometric identities,\n[\n\sin 45^\circ = \frac{\sqrt{2}}{2}\n]\nBut because $315^\circ$ is in the fourth quadrant where sine is negative:\n[\n\sin 315^\circ = -\frac{\sqrt{2}}{2}\n]", "---", "## Putting It All Together: The Complete Expression", "Putting all parts together, the original expression simplifies beautifully:\n[\n\sin 315^\circ = \sin(360^\circ - 45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2}\n]\nThis process highlights how a foundational identity like $\ an 45^\circ = 1$ supports deeper computations—grounding even complex angles in the geometry and symmetry of the unit circle.", "---", "## Why This Matters: Applications in Science and Engineering", "Understanding such trigonometric relationships allows professionals in physics, computer graphics, robotics, and signal processing to accurately model periodic phenomena, rotational motion, and wave behavior. Mastery of identities like $\sin(360^\circ - \ heta)$ ensures precise calculations and efficient problem-solving.", "---", "## Conclusion: From Basics to Broader Application", "The journey from $\ an 45^\circ = 1$ to computing $\sin 315^\circ = -\frac{\sqrt{2}}{2}$ demonstrates the elegance and interconnectedness of trigonometric principles. Whether solving academic problems or designing real-world systems, these identities form a reliable toolkit rooted in geometry and logic.", "If you're studying trigonometry or solving related problems, remembering these key steps ensures clarity and confidence—making the transition from simple to complex angles seamless.", "---", "Keywords: $\ an 45^\circ = 1$, $\sin 315^\circ$, trigonometry, unit circle, reference angle, negative sine, $\sin(360^\circ - \ heta)$, exact values, angular identities, math explanation, trigonometric identities.", "---", "Elevate your trigonometry skills today—start with the basics, and unlock the power of precise calculations across disciplines."]
