Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 25-36: Verify the reduction formula. $$ \sin (\theta+\pi)=-\sin \theta $$

Step-by-Step Solution

Answer

The reduction formula is verified as it simplifies to \(-\sin(\theta)\).

Step 1: Understanding the Equation

The problem asks us to verify the reduction formula for the sine function. The formula given is \(\sin(\theta + \pi) = -\sin(\theta)\). We will use trigonometric identities to prove this.

Step 2: Using Sine Addition Formula

The sine addition formula states that \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \). Applying this to \( \theta + \pi \), we get:\[\sin(\theta + \pi) = \sin(\theta)\cos(\pi) + \cos(\theta)\sin(\pi)\]

Step 3: Substitute Known Values

From trigonometric values, we know that \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\). Substituting these values into the equation, we have:\[\sin(\theta + \pi) = \sin(\theta)(-1) + \cos(\theta)(0)\]

Step 4: Simplify the Equation

Simplify the expression by multiplying:\[\sin(\theta + \pi) = -\sin(\theta)\]The term with \(\cos(\theta)\cdot 0\) becomes 0, confirming the formula.

Step 5: Conclusion

The reduction formula \(\sin(\theta + \pi) = -\sin(\theta)\) is verified using the trigonometric identities for sine addition and known values for \(\pi\).

Key Concepts

Reduction Formula

The reduction formula is used to modify trigonometric functions by shifting their angles. It helps simplify complex expressions or solve trigonometric equations. By reducing an angle to a more familiar or simpler one, calculations become more straightforward.

• In our case, the reduction formula under discussion simplifies the sine of an angle incremented by \( \pi \).
• The formula states that \( \sin(\theta + \pi) = -\sin(\theta) \).
• This means that adding \( \pi \) to \( \theta \) reflects the sine value over the origin.

Reduction formulas are essential for trigonometric analysis, particularly for verifying identities or transforming functions into simpler forms.
They rely on known trigonometric properties and are fundamental in calculus and physics.

Sine Function

The sine function is a fundamental trigonometric function. It is defined in the context of a right-angled triangle as the ratio of the length of the opposite side to the hypotenuse. In the unit circle, it corresponds to the y-coordinate of a point.

• The sine of an angle \( \theta \) can be expressed as \( \sin(\theta) \).
• It is periodic with a period of \( 2\pi \), meaning that \( \sin(\theta + 2\pi) = \sin(\theta) \).
• It is an odd function, hence \( \sin(-\theta) = -\sin(\theta) \).

Understanding the sine function is crucial for many areas of mathematics and physics, such as wave physics, signal processing, and oscillating motion analysis.
Its behavior under transformations like shifts and reflections is key to solving trigonometric problems.

Addition Formula

The addition formula is a trigonometric identity used to find the sine or cosine of the sum of two angles. It breaks down complex angles into simpler components that are easier to handle.

• The addition formula for sine is \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \).
• This formula allows us to decompose the problem into familiar parts, using known sine and cosine values.
• For our specific problem, substituting \( \theta \) and \( \pi \) provides a way to verify the reduction formula.

Applying the addition formula is a powerful technique when dealing with angle transformations, particularly in calculus where angle manipulations are common.
Understanding how to apply it facilitates solving more complex trigonometric identities.

Trigonometric Values

Trigonometric values like \( \sin(\pi) \) and \( \cos(\pi) \) are constants that accurately describe the function's results at key angles. These values are crucial for simplifying expressions and proving identities.

• The value of \( \cos(\pi) \) is \(-1\) because \( \pi \) radians correspond to 180 degrees, making a full half-circle on the unit circle.
• Similarly, \( \sin(\pi) \) is \(0\), reflecting the y-coordinate at this angle.
• Using these values, specific expressions such as \( \sin(\theta + \pi) \) can be quickly simplified and verified.

Knowing these trigonometric values by heart aids in computing complex expressions more rapidly.
They serve as building blocks for more complicated calculations in mathematics and its applications.