Problem 48 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Derive the formula \(\sin A x \sin B x=\frac{1}{2}[\cos (A-B) x-\cos (A+B) x]\) given in this section. (Begin with the addition formula for cosine.)

Step-by-Step Solution

Answer

To derive the formula \(\sin A x \sin B x=\frac{1}{2}[\cos (A-B) x-\cos (A+B) x]\), start with the addition and subtraction formulas for cosine. Next, express \(\sin A x \sin B x\) in terms of Cosine and substitute \(a=A x\) and \(b=B x\). This leads to the required formula.

Step 1: Identify the Cosine addition and subtraction formulas

The addition and subtraction formulas for cosine are: \(\cos (a+b) = \cos (a) \cos (b) - \sin(a) \sin (b)\) and \(\cos (a-b) = \cos(a) \cos(b) + \sin(a) \sin(b)\). You need these to derive the expression.

Step 2: Express \(\sin A x \sin B x\) in terms of Cosine

To express \(\sin A x \sin B x\) in terms of Cosine, subtract \(\cos (a-b)\) from \(\cos(a+b)\), from the aforementioned identities, then you'll get: \(\cos (a+b) - \cos (a-b) = 2 \sin(a) \sin(b)\).

Step 3: Substitute expressions

Substitute \(a=A x\) and \(b=B x\), and divide both sides by 2. So, it follows that \(\sin A x \sin B x = \frac{1}{2}[\cos (A-B) x-\cos (A+B) x]\).

Step 4: Conclude the Proof

The formula \(\sin A x \sin B x=\frac{1}{2}[\cos (A-B) x-\cos (A+B) x]\) has now been derived successfully from the addition and subtraction formula for cosine.

Key Concepts

Cosine Addition Formulas

The cosine addition formulas are a cornerstone of trigonometry. They express the cosine of the sum or difference of two angles in terms of the sines and cosines of the angles themselves. For instance, the formula for the cosine of a sum is
\[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \],
and for the difference, we have
\[ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \].
These formulas allow us to perform complex angle calculations, including the derivation of other important trigonometric identities. For example, by subtracting these two equations, we arrive at an expression for the product of sines:
\[ \cos(a + b) - \cos(a - b) = -2 \sin(a)\sin(b) \].
After dividing by -2, we get
\[ \sin(a)\sin(b) = \frac{-1}{2}[\cos(a + b) - \cos(a - b)] \].
Understanding the cosine addition formulas is not only necessary for solving trigonometric problems but also forms the basis for more advanced mathematical concepts in fields such as calculus and physics.

Sine Function Properties

The sine function, denoted as \( \sin \), exhibits various fundamental properties that are extensively used in trigonometry. Among these are its periodicity, symmetry, and relationship to the unit circle.

Periodicity

The function is periodic with a period of \( 2\pi \) radians or 360 degrees. This means that \( \sin(\theta + 2\pi) = \sin(\theta) \), essentially repeating its values over intervals of \( 2\pi \).

Odd Function Symmetry

The sine function is also an odd function. This implies that \( \sin(-\theta) = -\sin(\theta) \), reflecting the function's symmetry about the origin in the coordinate plane.

Relation to Unit Circle

Furthermore, the sine of an angle in the unit circle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. This geometric representation helps us understand why sine values oscillate between -1 and 1 and underpins the meaning of sine in the context of triangles and oscillatory motions.

All these properties, combined with trigonometric identities, allow us to solve complex problems and analyze periodic phenomena in various scientific fields.

Subtraction Formulas in Trigonometry

Subtraction formulas in trigonometry are just as vital as addition formulas, and they all relate closely to one another. The subtraction formulas can be seen as an extension of the addition formulas and are particularly helpful for expressing trigonometric functions of a difference between two angles in terms of functions of individual angles.

For cosine, we've seen that
\[ \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \].
In parallel, we have a similar formula for sine:
\[ \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \].
These formulas facilitate the transformation and simplification of trigonometric expressions. They also play an essential role in proving various other identities and solving equations.

For instance, the exercise given illustrates a case where subtraction formulas can simplify the expression for the product of two sine functions into a more manageable form, which then aids in the calculation and understanding of wave interference, signal processing, and acoustics in physics.