SAT Question e3158182: Right Triangles And Trigonometry answer and explanation with Desmos solution

Choice D is correct. It’s given that acute angle $P$ has a measure of $p$ radians and that acute angle $T$ has a measure of $t$ radians. If $\sin p=\cos t$, it follows that $p+t=\frac{\pi }{2}$, or $p=\frac{\pi }{2}-t$. It’s also given that function $g$ is defined by $g(x)=p(8{)}^x+6t$. To determine the value of $g(0)$ in terms of $t$, substituting $0$ for $x$ and $\frac{\pi }{2}-t$ for $p$ in the equation $g(x)=p(8{)}^x+6t$ yields $g(0)=\left(\frac{\pi }{2}-t\right)(8{)}^0+6t$, or $g(0)=\left(\frac{\pi }{2}-t\right)(1)+6t$. This is equivalent to $g(0)=\frac{\pi }{2}-t+6t$, or $g(0)=5t+\frac{\pi }{2}$. Therefore, $5t+\frac{\pi }{2}$ represents the value of $g(0)$ in terms of $t$.