Problem 35 - Calculus Single Variable

Question

The equations \(\sin (2 \theta)=\sin (\theta) \cos (\theta)\) and \(\sin (2 \theta)=\) \(2 \sin (\theta) \cos (\theta)\) are fundamentally different. Explain why.

Step-by-Step Solution

Answer

The equations differ because the first lacks the "2" from the sine double angle identity.

Step 1: Understand the double angle formula for sine

The double angle identity for sine states that \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\). This formula expresses the sine of a double angle in terms of the sine and cosine of the original angle \(\theta\).

Step 2: Compare given equations to standard identity

The first equation provided, \(\sin (2 \theta)=\sin (\theta) \cos (\theta)\), does not match the standard identity for \(\sin(2\theta)\), which should be \(2 \sin(\theta) \cos(\theta)\).

Step 3: Analyze the structure of each equation

In the first equation \(\sin(2\theta) = \sin(\theta) \cos(\theta)\), the coefficient "2" is missing, while the second equation \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\) includes the correct coefficient from the double angle identity.

Step 4: Identify the differences

The fundamental difference between the two equations is the presence of the coefficient "2" in the double angle identity. The first equation is not the correct formulation of the sine double angle identity as it is missing this factor.

Key Concepts

Double Angle Identity

The double angle identity is a fundamental concept in trigonometry which helps to simplify expressions involving trigonometric functions of double angles. Specifically, for the sine function, the double angle identity is expressed as:\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]This identity is highly useful because it allows us to express the sine of a double angle, \(2\theta\), in terms of simpler functions - the sine and cosine of \(\theta\). This simplification has various applications in solving trigonometric equations, integrating trigonometric functions, and analyzing wave forms, among other uses.
When using the double angle identity, remember:

• "\(2 \sin(\theta) \cos(\theta)\)" expresses the product of twice the sine and cosine of the angle \(\theta\).
• It is derived from the sum of angle identities for sine in trigonometry.
• It essential for rewriting functions in terms of single angles for easier computations.

Sine Function

The sine function is one of the basic trigonometric functions and is defined for angles within the context of a right-angled triangle and the unit circle.Its basic definition is related to the ratio of the side opposite the angle to the hypotenuse in a right triangle:

• In Right Triangle: \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• On Unit Circle: The sine of an angle \(\theta\) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the unit circle.

The sine function is periodic with a period of \(2\pi\), which means its values repeat every \(2\pi\) radians.The double angle identity is particularly important as it transforms double angle calculations into basic sine and cosine operations. In other contexts, the sine function is key in describing oscillatory motions like sound and light waves.
Key characteristics of the sine function:

• It has a range from -1 to 1.
• The function is a smooth, continuous wave.
• Sine values are zero at integer multiples of \(\pi\).

Equation Comparison

Comparing equations involves looking at their structure and coefficients to understand their implications better. In this scenario, we are comparing two equations related to trigonometric identities.Let's go over how to identify the differences between these two:1. The provided equations: - \(\sin (2 \theta)=\sin (\theta) \cos (\theta)\) - \(\sin (2 \theta)=2 \sin (\theta) \cos (\theta)\)2. Standard Identity Check: - The standard double angle identity for sine is \(2 \sin(\theta) \cos(\theta)\). - This identity requires the multiplication of sine and cosine by 2 because that represents the essential structure of the sine double angle.
3. Differences Noted: - The first equation lacks the crucial multiplier of 2, which makes it fundamentally incorrect according to the double angle identity. - The absence of "2" in the first equation affects not just aesthetic correctness but also practical calculations depending on this identity.
Comparison helps to ensure mathematical equations align correctly with standard identities, which is crucial for solving equations correctly and maintaining the coherence of mathematical frameworks.