use the figure to evaluate the following function, given that f(x)=sin x. f(2θ) f(2θ)= (simplify your…

Explanation:

Step1: Find the value of (a)

Since the point ((a,2)) lies on the circle (x^{2}+y^{2}=5), substitute (x = a) and (y = 2) into the equation: (a^{2}+2^{2}=5), so (a^{2}=5 - 4=1), then (a=\pm1). From the position of the point in the first - quadrant, (a = 1).

Step2: Find (\sin\theta) and (\cos\theta)

We know that for a point ((x,y)) on the unit - circle related to an angle (\theta), (\sin\theta=\frac{y}{r}) and (\cos\theta=\frac{x}{r}). Here (r=\sqrt{5}), (x = 1), (y = 2), so (\sin\theta=\frac{2}{\sqrt{5}}) and (\cos\theta=\frac{1}{\sqrt{5}}).

Step3: Use the double - angle formula for sine

The double - angle formula for sine is (\sin(2\theta)=2\sin\theta\cos\theta). Substitute (\sin\theta=\frac{2}{\sqrt{5}}) and (\cos\theta=\frac{1}{\sqrt{5}}) into the formula: (\sin(2\theta)=2\times\frac{2}{\sqrt{5}}\times\frac{1}{\sqrt{5}}).

Step4: Simplify the expression

(\sin(2\theta)=\frac{4}{5}).

Answer:

(\frac{4}{5})