Problem 57 - Precalculus Mathematics for Calculus

Question

Write the product as a sum. $$\cos x \sin 4 x$$

Step-by-Step Solution

Answer

\(\cos x \sin 4x = \frac{1}{2}(\sin 5x + \sin 3x)\).

Step 1: Identify the Trigonometric Identity to Use

The expression involves a product of sine and cosine functions. The product-to-sum identities are appropriate here. Specifically, we can use the identity for the product of cosine and sine: \[\cos A \sin B = \frac{1}{2} \left( \sin(A + B) - \sin(A - B) \right).\]

Step 2: Assign Values to A and B

Identify the values of \(A\) and \(B\) in the expression \(\cos x \sin 4x\). Here, \(A = x\) and \(B = 4x\).

Step 3: Apply the Identity

Substitute \(A = x\) and \(B = 4x\) into the identity: \[\cos x \sin 4x = \frac{1}{2} \left( \sin(x + 4x) - \sin(x - 4x) \right).\]Simplify the expression inside the sine functions:\[= \frac{1}{2} \left( \sin 5x - \sin(-3x) \right).\]

Step 4: Simplify the Expression

Recall that \(\sin(-\theta) = -\sin(\theta)\). Thus, the expression becomes:\[= \frac{1}{2} \left( \sin 5x + \sin 3x \right).\]

Step 5: Finalize the Expression

The product \(\cos x \sin 4x\) can be written as a sum:\[\frac{1}{2}(\sin 5x + \sin 3x).\]

Key Concepts

Product-to-Sum Identities

Product-to-sum identities are part of the family of trigonometric identities that transform products of trigonometric functions into sums or differences. These are especially useful because adding and subtracting functions is often simpler than multiplying them, especially when solving integrals or simplifying expressions.
These identities are derived from the angle addition and subtraction identities for sine and cosine. By converting the product into a sum or difference, they facilitate operations in calculus and trigonometry.
Here are the identities commonly used:

• For sine and cosine: \(\cos A \sin B = \frac{1}{2} \left( \sin(A + B) - \sin(A - B) \right)\)
• For the product of two sines: \(\sin A \sin B = \frac{1}{2} \left( \cos(A - B) - \cos(A + B) \right)\)
• For the product of two cosines: \(\cos A \cos B = \frac{1}{2} \left( \cos(A + B) + \cos(A - B) \right)\)

Understanding these can significantly help when working with trigonometric expressions, whether you are evaluating an integral, simplifying a complex expression, or solving equations.

Sine Function

The sine function is one of the fundamental trigonometric functions, alongside cosine and tangent. It is defined in the context of a right triangle, as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. More generally, it can be defined using the unit circle.
The function is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) radians:

• It ranges between -1 and 1.
• Sine is an odd function, which implies that \(\sin(-\theta) = -\sin(\theta)\).
• It starts at 0, increases to 1 by \(\frac{\pi}{2}\), returns to 0 at \(\pi\), decreases to -1 at \(\frac{3\pi}{2}\), and completes the cycle at \(2\pi\).

In the context of product-to-sum identities, understanding the sine function's properties helps simplify expressions, particularly in switching negative sine values to positive when manipulating terms as seen in the identity \(\cos x \sin 4x = \frac{1}{2} \left( \sin 5x - \sin(-3x) \right)\).

Cosine Function

The cosine function is another fundamental trigonometric function associated with a right triangle. It is the ratio of the length of the adjacent side to an angle over the hypotenuse's length in a right triangle. Extended to the unit circle, cosine represents the x-coordinate of a point on the circle.
Like the sine function,

• Cosine is periodic with a period of \(2\pi\).
• Its range lies between -1 and 1.
• Unlike sine, cosine is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• It starts at 1, decreases to 0 at \(\frac{\pi}{2}\), hits -1 at \(\pi\), returns to 0 at \(\frac{3\pi}{2}\), and completes a cycle back at 1 by \(2\pi\).

Applying the cosine function in product-to-sum identities aids in transforming the products of sine and cosine into manageable forms, such as the expression \(\cos x \sin 4x\). This transformation makes solving or simplifying trigonometric problems less cumbersome.