["# Compute $\ an 45^\circ + \sin 315^\circ$ – A Step-by-Step Breakdown", "Understanding basic trigonometric functions is essential for students, educators, and anyone working with angles. One common problem students encounter is computing expressions like $\ an 45^\circ + \sin 315^\circ$. This article guides you step-by-step through evaluating this expression using standard trigonometric values, offering clarity on how to compute angular functions across different quadrants. Whether you're preparing for exams or just want to reinforce your math skills, this guide breaks down the computation clearly and precisely.", "## Step 1: Recall the Definitions", "To evaluate $\ an 45^\circ + \sin 315^\circ$, we first recall what tangent and sine mean in the unit circle:
\n- $\ an \ heta = \frac{\sin \ heta}{\cos \ heta}$
\n- $\sin \ heta$ gives the vertical coordinate (y-value) of a point on the unit circle corresponding to angle $\ heta$.", "Angles are measured from the positive x-axis, increasing counterclockwise. Recognizing where $45^\circ$ and $315^\circ$ lie helps determine their coordinates and perform accurate computations.", "## Step 2: Compute $\ an 45^\circ$", "The angle $45^\circ$ lies in the first quadrant, where all trigonometric functions are positive. From standard trigonometric values:
\n$$
\n\ an 45^\circ = 1
\n$$
\nThis value is widely known because $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$, so their ratio simplifies neatly to 1.", "## Step 3: Compute $\sin 315^\circ$", "The angle $315^\circ$ lies in the fourth quadrant, where sine values are negative. To find its sine, we note:
\n- $315^\circ = 360^\circ - 45^\circ$, so it forms a reference angle of $45^\circ$.
\n- In the fourth quadrant, $\sin(360^\circ - \ heta) = -\sin \ heta$, so:
\n$$
\n\sin 315^\circ = \sin(360^\circ - 45^\circ) = -\sin 45^\circ = -\frac{\sqrt{2}}{2}
\n$$", "## Step 4: Add the Values", "Now substitute the computed values into the original expression:
\n$$
\n\ an 45^\circ + \sin 315^\circ = 1 + \left(-\frac{\sqrt{2}}{2}\right) = 1 - \frac{\sqrt{2}}{2}
\n$$", "This expression combines a rational number and an irrational term, representing an exact algebraic form. For decimal approximation:
\n$$
\n1 - \frac{\sqrt{2}}{2} \approx 1 - 0.7071 = 0.2929
\n$$", "## Step 5: Final Thought", "The sum $\ an 45^\circ + \sin 315^\circ = 1 - \frac{\sqrt{2}}{2}$ illustrates how trigonometric identities and quadrant knowledge simplify complex expressions. Accurately knowing reference angles and sign rules enhances efficiency. This problem reinforces foundational skills vital for advanced trigonometry, engineering, physics, and more.", "### Summary", "$$
\n\ an 45^\circ + \sin 315^\circ = 1 + \left(-\frac{\sqrt{2}}{2}\right) = 1 - \frac{\sqrt{2}}{2} \approx 0.2929
\n$$", "Understanding how to compute such sums empowers learners to tackle a wide range of mathematical challenges confidently and precisely."]
