91Ó°ÊÓ

Cross multiplication is a method used to solve equations involving fractions. It helps us eliminate the fractions to make the equation easier to solve. In the given problem: \(\frac{\sin 60^{\circ}}{6}=\frac{\sin 10^{\circ}}{a}\), we use cross multiplication to get rid of the denominators.Here's a simple way to apply cross multiplication:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Do the same for the other pair (numerator of the second by the denominator of the first).

This means we multiply \(\sin 60^{\circ}\) by \(a\) and \(\sin 10^{\circ}\) by 6, which results in: \[a \sin 60^{\circ} = 6 \sin 10^{\circ}\]After cross-multiplying, we have an equation without fractions, making it easier to isolate and solve for the unknown variable (\(a\)). It’s a very handy technique, especially when dealing with proportionate relationships.

The sine function is a fundamental trigonometric function. It is used to describe a relationship between the angles and sides of right-angled triangles. For an angle \(\theta\), the sine function, written as \(\sin \theta\), is defined as: \[\sin \theta = \frac{{\text{opposite side}}}{{\text{hypotenuse}}}\]In our problem, we need to find the sine values for specific angles:

• \(\sin 60^{\circ} = 0.866\)
• \(\sin 10^{\circ} = 0.174\)

These values can be obtained using a scientific calculator or trigonometric tables. Once we have the sine values, we substitute them back into the equation to continue solving for our variable \(a\). Understanding these values is crucial, as they represent the ratio we need to find our solution.

Rounding numbers is the process of simplifying a number to make it easier to work with, often to a specified degree of precision. In our exercise, we need to round the final answer to the nearest tenth. Here's a simple guide on how to round numbers:

• Identify the digit in the place you are rounding to (the tenths place in this case).
• Look at the digit immediately to the right of this place (the hundredths place).
• If this digit is 5 or greater, increase the digit in the tenths place by one.
• If it's less than 5, leave the digit in the tenths place unchanged.

For our solution, calculating \(\frac{1.044}{0.866}\) gives us approximately 1.2. Since the hundredths place is less than 5, we do not change the tenths place. Therefore, our final answer, 1.2, is already rounded to the nearest tenth. Rounding helps make numbers more manageable, especially in practical situations or when communicating results.