Question

find the exact value of \\( \sin a \\) in simplest radical form.

Explanation:

Step1: Recall the definition of sine in a right triangle

In a right triangle, the sine of an acute angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For angle \( A \) in right triangle \( ABC \) (right - angled at \( C \)), \(\sin A=\frac{\text{opposite side to }A}{\text{hypotenuse}}\).

Step2: Identify the opposite side and hypotenuse

• The side opposite to angle \( A \) is \( BC \), and from the diagram, \( BC = 3 \).
• The hypotenuse of the right triangle is \( AB \), and from the diagram, \( AB=9 \).

Step3: Calculate \(\sin A\)

Using the definition of sine, \(\sin A=\frac{BC}{AB}\). Substituting the values of \( BC = 3 \) and \( AB = 9 \), we get \(\sin A=\frac{3}{9}=\frac{1}{3}\). Wait, but let's check if the triangle is a right triangle. Let's verify using the Pythagorean theorem. \( AC=\sqrt{72}\), \( BC = 3 \), \( AB = 9 \).

We know that in a right triangle, \( AC^{2}+BC^{2}\) should be equal to \( AB^{2}\) (by Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c \) is the hypotenuse and \( a,b \) are the legs).

Calculate \( AC^{2}+BC^{2}\): \( (\sqrt{72})^{2}+3^{2}=72 + 9=81 \), and \( AB^{2}=9^{2} = 81 \). So the triangle is a right triangle, and our identification of opposite side and hypotenuse is correct.

Answer:

\(\frac{1}{3}\)