Question

find t.
triangle image with right angle, 3√2 in side, 60° angle, 30° angle, and t as a side
write your answer in simplest radical form.
blank inches

Explanation:

Step1: Identify trigonometric relation

In a right - triangle, we can use the tangent function. For an angle \(\theta\) in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, for the \(60^{\circ}\) angle, the adjacent side to \(60^{\circ}\) is \(3\sqrt{2}\) and the opposite side is \(t\). We know that \(\tan60^{\circ}=\sqrt{3}\). Also, we can use the fact that \(\tan60^{\circ}=\frac{t}{3\sqrt{2}}\) (since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), where \(\theta = 60^{\circ}\), opposite side is \(t\) and adjacent side is \(3\sqrt{2}\)).

Step2: Solve for \(t\)

We know that \(\tan60^{\circ}=\sqrt{3}\), and from the tangent formula \(\tan60^{\circ}=\frac{t}{3\sqrt{2}}\). So we can set up the equation \(\sqrt{3}=\frac{t}{3\sqrt{2}}\).
To solve for \(t\), we multiply both sides of the equation by \(3\sqrt{2}\):
\(t = 3\sqrt{2}\times\sqrt{3}\)
Using the property of radicals \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) (\(a = 2\), \(b = 3\)), we get \(t=3\sqrt{6}\)

Answer:

\(3\sqrt{6}\)