1. The problem involves understanding the function $f(a)$ defined as: $$ f(a) = \begin{cases} 0 & \text{for } 0 \leq a < a_1 \\ 0.5 & \text{for } a = a_1 \\ 1 & \text{for } a_1 < a \leq 90^\circ \end{cases} $$ 2. The graph shows $f(a)$ on the y-axis from 0 to 1, and $a$ on the x-axis from 0 to 100, with a vertical line at $a_1 \approx 50$. 3. The function $f(a)$ is a step function that jumps from 0 to 0.5 at $a = a_1$, then immediately to 1 for $a > a_1$. 4. The problem also references the trigonometric identity: $$\cos \theta = \sin \alpha \sin \beta + \cos \alpha \cos \theta$$ which appears to be a variation of the cosine addition formula, but the exact problem is to analyze $f(a)$. 5. The key is to interpret $f(a)$ as a piecewise function with a jump discontinuity at $a_1$. 6. The value of $a_1$ is approximately 50 on the x-axis, where $f(a)$ jumps from 0 to 0.5, then to 1. 7. The function $f(a)$ can be written as: $$ f(a) = \begin{cases} 0 & a < 50 \\ 0.5 & a = 50 \\ 1 & a > 50 \end{cases} $$ 8. This means for any $a$ less than 50, $f(a) = 0$; at $a=50$, $f(a) = 0.5$; and for $a$ greater than 50, $f(a) = 1$. 9. This step function models a sudden change at $a_1 = 50$. Final answer: $$ f(a) = \begin{cases} 0 & 0 \leq a < 50 \\ 0.5 & a = 50 \\ 1 & 50 < a \leq 90^\circ \end{cases} $$