Question

find the missing lengths of the given triangle. give an exact answer and, where appropriate, an approximation to the nearest tenth. a is equal to \\(\square\\) units. (type an integer or a decimal )

Explanation:

Step1: Use trigonometric ratios

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 60^{\circ}\), the hypotenuse \(c = 9\), and the side opposite to the \(60^{\circ}\) angle is \(a\). So, \(\sin60^{\circ}=\frac{a}{9}\).
Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{a}{9}\).

Step2: Solve for \(a\)

Cross - multiply: \(a = 9\times\frac{\sqrt{3}}{2}\).
The exact value of \(a=\frac{9\sqrt{3}}{2}\).
For the approximation, \(\sqrt{3}\approx1.732\). Then \(a=\frac{9\times1.732}{2}=\frac{15.588}{2}=7.794\approx7.8\).

Answer:

The exact value of \(a\) is \(\frac{9\sqrt{3}}{2}\) units and the approximate value (to the nearest tenth) is \(7.8\) units.