Problem 55 - Algebra and Trigonometry

Question

\(55-60\) Write the product as a sum. $$ \sin 2 x \cos 3 x $$

Step-by-Step Solution

Answer

\(\sin 2x \cos 3x = \frac{1}{2} [\sin(5x) - \sin(x)]\)

Step 1: Identify the Trigonometric Product-to-Sum Formula

To convert the product of trigonometric functions into a sum, we use the identity for the product of sine and cosine:\[ \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \]For this problem, identify that \(A = 2x\) and \(B = 3x\).

Step 2: Substitute into the Formula

Use the identified values of \(A = 2x\) and \(B = 3x\) in the product-to-sum formula:\[ \sin 2x \cos 3x = \frac{1}{2} [\sin(2x + 3x) + \sin(2x - 3x)] \]

Step 3: Simplify the Sums Inside the Sine

Simplify the expressions inside the sine functions:- \(2x + 3x = 5x\)- \(2x - 3x = -x\)Substitute these back into the expression:\[ \sin 2x \cos 3x = \frac{1}{2} [\sin(5x) + \sin(-x)] \]

Step 4: Use the Odd Property of Sine

Recall that sine is an odd function, meaning \(\sin(-x) = -\sin(x)\). Apply this property:\[ \sin 2x \cos 3x = \frac{1}{2} [\sin(5x) - \sin(x)] \]

Key Concepts

Product-to-Sum Formulas

When dealing with trigonometric functions, you may often encounter situations where you need to convert products into sums. This is facilitated by the use of **product-to-sum formulas**. These formulas are handy tools in trigonometry, especially useful when integrating trigonometric functions or simplifying expressions.

For instance, let's take the product \[\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]\] This tells us that the product of sine and cosine can be expressed as a sum of sines. The application of these formulas helps to express the interactions between angles in a more manageable way.

In solving problems, remember:

• Determine the exact angles for \(A\) and \(B\). Assess what you are given and match it to the appropriate formula.
• Substitute the values for each angle into the product-to-sum formula.

These strategies ensure you're able to convert complex trigonometric products into simpler sum expressions.

Sine Function

The sine function is one of the fundamental building blocks in trigonometry. It relates to the y-coordinate of a point on the unit circle as an angle, \(\theta\), is measured from the positive x-axis.

The sine function is periodic, with a period of \(2\pi\), which means its values repeat every \(2\pi\) radians. It's also important to note that sine is an **odd function**, meaning:\[\sin(-x) = -\sin(x)\]This property is crucial when simplifying expressions like \(\sin(-x)\) found in product-to-sum conversions.

Some key concepts about the sine function:

• It ranges between -1 and 1.
• It has a maximum point at \(\frac{\pi}{2} + 2k\pi\) and a minimum at \(\frac{3\pi}{2} + 2k\pi\) for integer \(k\).
• The sine of an angle represents the vertical component of a point on the unit circle.

Mastering the sine function helps in understanding wave patterns and various applications in physics and engineering.

Cosine Function

The cosine function works alongside sine as a fundamental trigonometric function. It describes the x-coordinate of a point on the unit circle, providing a perspective on how angles open relative to the horizontal axis.

Cosine is a **even function**, distinctly different from sine, as:\[\cos(-x) = \cos(x)\]This means the cosine of an angle is the same even if the angle is negated.

The cosine function also has a period of \(2\pi\), just like the sine function, reflecting its recurring pattern over a full circle.

Key aspects of cosine include:

• The range is similar to that of sine, between -1 and 1.
• Cosine peaks at 1 and has troughs at -1.
  These points occur at integer multiples of \(\pi\).
• In right-angled triangles, it represents the ratio of the adjacent side to the hypotenuse.

By understanding the cosine function, you gain insight into rotations, oscillations, and many phenomena in both math and science.