Problem 84 - Algebra and Trigonometry

Question

Use words to describe the formula for each of the following: the sine of the difference of two angles.

Step-by-Step Solution

Answer

The sine of the difference of two angles is given by the formula \(\sin(a - b) = \sin a \cos b - \cos a \sin b\), where \(\sin a \cos b\) represents the product of the sine of angle \(a\) and the cosine of angle \(b\), and \(\cos a \sin b\) represents the product of the cosine of angle \(a\) and the sine of angle \(b\). These two quantities are then subtracted from each other to arrive at the result.

Step 1 - Recall the formula

The formula for the sine of the difference of two angles (\(a\) and \(b\)) is given by: \(\sin(a - b) = \sin a \cos b - \cos a \sin b\). This is one of many trigonometric identities that can be derived using the definitions of sine and cosine.

Step 2 - Explain parts of the formula

In this formula, \(\sin a \cos b\) represents multiplying the sine of angle \(a\) and cosine of angle \(b\). The next part, \(\cos a \sin b\), is multiplying the cosine of angle \(a\) and the sine of angle \(b\). These two parts are then subtracted from each other to get the sine of the difference of the two angles.

Step 3 - The importance of the formula

Understanding this formula is crucial as it allows you to find the sine of the difference of two angles without needing to know the exact measurements of the angles. This can be very helpful in many areas of mathematics and physics.