Problem 75 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the product-to-sum formulas to write the product as a sum or difference. $$6 \sin \frac{\pi}{3} \cos \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The product \(6 \sin \frac{\pi}{3} \cos \frac{\pi}{3}\) can be written as a sum by using the product-to-sum identity, turning into \(3 \sin \frac{2\pi}{3}\).

Step 1: Identify a and b in the equation

In this case, both \(a\) and \(b\) are both \(\frac{\pi}{3}\) in the given equation \(6 \sin \frac{\pi}{3} \cos \frac{\pi}{3}\).

Step 2: Rewrite the product as a sum using the product-to-sum identity

The problem can be rewritten using the formula \(2 \sin a \cos b = \sin(2a)\). Hence, substituting \(a = \frac{\pi}{3}\) and, \(2a = 2*\frac{\pi}{3} = \frac{2\pi}{3}\), and \(b = \frac{\pi}{3}\), we get \(6 \sin \frac{\pi}{3} \cos \frac{\pi}{3} = 6*\frac{1}{2} \sin \frac{2\pi}{3}\)

Step 3: Simplify and final solution

Upon simplification, you yield \(6*\frac{1}{2} \sin \frac{2\pi}{3} = 3 \sin \frac{2\pi}{3}\) which is our desired rewritten product in terms of a sum or difference

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that relate the angles and sides of triangles. These identities are incredibly useful for simplifying expressions and solving equations involving trigonometric functions like sine, cosine, and tangent.
One such identity that's particularly handy is the product-to-sum formula. This formula helps in rewriting products of sine and cosine functions as sums or differences, making them easier to work with. Specifically, for the product of sine and cosine functions, we have the identity:

• \( 2 \sin a \cos b = \sin(a+b) + \sin(a-b) \)
• \( 2 \cos a \sin b = \sin(a+b) - \sin(a-b) \)

These identities simplify complex trigonometric expressions into more manageable sums or differences. This allows for easier and more efficient computation in calculus and other advanced mathematical operations.

Sine and Cosine Functions

The sine and cosine functions, often written as \( \sin(\theta) \) and \( \cos(\theta) \), describe the relationship between the angles and ratios of right-angled triangles. They are periodic functions that oscillate, which means they repeat their values in regular intervals.
The core definitions are based on the unit circle:

• **Sine function**: defines the y-coordinate of a point on the unit circle. It's periodic with a period of \(2\pi\), meaning \( \sin(\theta + 2\pi) = \sin(\theta) \).
• **Cosine function**: defines the x-coordinate of a point on the unit circle. It shares the same periodicity as sine with \(2\pi\), \( \cos(\theta + 2\pi) = \cos(\theta) \).

These functions are not only essential in geometry, but they also play critical roles in physics, engineering, and other fields involving wave patterns and oscillations.

Angle Conversion

Converting angles between degrees and radians is crucial in trigonometry since different contexts use different units. Radians are a natural measurement system for angles, used most commonly in calculus and mathematics.

• **Degrees to Radians**: To convert degrees to radians, multiply by \( \frac{\pi}{180} \). For instance, 60 degrees is \(60 \times \frac{\pi}{180} = \frac{\pi}{3}\).
• **Radians to Degrees**: To convert radians to degrees, multiply by \( \frac{180}{\pi} \). For example, \( \frac{\pi}{3} \) radians converts to \(\frac{\pi}{3} \times \frac{180}{\pi} = 60 \text{ degrees}\).

Understanding these conversions is essential for solving trigonometric equations accurately, especially when the measurements involve integrating with real-world situations, where angles might be initially provided in degrees.