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 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, \( BC = 5 \).
• The hypotenuse of the right triangle is \( AB \), and from the diagram, \( AB=10 \).

Step3: Calculate \(\sin A\)

Using the definition of sine, \(\sin A=\frac{BC}{AB}\). Substituting the values of \( BC = 5 \) and \( AB = 10 \), we get \(\sin A=\frac{5}{10}=\frac{1}{2}\). Wait, but let's check the triangle with the given side \( AC=\sqrt{75}\). Let's verify using the Pythagorean theorem. In a right triangle, \( AC^{2}+BC^{2}=AB^{2}\).

Calculate \( AC^{2}+BC^{2} \): \((\sqrt{75})^{2}+5^{2}=75 + 25=100\), and \( AB^{2}=10^{2} = 100 \). So the triangle is valid. But wait, the opposite side to angle \( A \) is \( BC \), hypotenuse is \( AB \). So \(\sin A=\frac{BC}{AB}=\frac{5}{10}=\frac{1}{2}\)? Wait, no, wait. Wait, angle \( A \): in triangle \( ABC \), right - angled at \( C \), angle \( A \) is at vertex \( A \), so the sides: adjacent to \( A \) is \( AC \), opposite is \( BC \), hypotenuse is \( AB \). So \(\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AB}\). Since \( BC = 5 \) and \( AB = 10 \), \(\sin A=\frac{5}{10}=\frac{1}{2}\). But let's re - check. Wait, \( \sqrt{75}=5\sqrt{3}\approx8.66 \), \( 5^{2}+(5\sqrt{3})^{2}=25 + 75 = 100=10^{2} \), so the triangle is a 30 - 60 - 90 triangle? Wait, no, if \( BC = 5 \), \( AB = 10 \), then the angle opposite to \( BC \) (angle \( A \)) has \(\sin A=\frac{5}{10}=\frac{1}{2}\), so angle \( A = 30^{\circ}\), which is correct because in a right triangle, if the side opposite 30 degrees is half the hypotenuse.

Wait, maybe I made a mistake in identifying the opposite side. Wait, angle \( A \): the sides: \( AC \) is adjacent, \( BC \) is opposite, \( AB \) is hypotenuse. So \(\sin A=\frac{BC}{AB}=\frac{5}{10}=\frac{1}{2}\). But let's check again. The formula for sine is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So for angle \( A \), opposite side is \( BC = 5 \), hypotenuse \( AB = 10 \), so \(\sin A=\frac{5}{10}=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)