Problem 96 - Precalculus with Limits

Question

In Exercises 95 - 98, verify the identity. \( \sin(n\pi + \theta) = \left(-1\right)^n \sin \theta \), \( n \) is an integer.

Step-by-Step Solution

Answer

The identity \( \sin(n\pi + \theta) = \left(-1\right)^n \sin \theta \), where \( n \) is an integer, is correctly verified.

Step 1: Expand the sine function on the left-hand side

Use the trigonometric identity of sine of sum \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) to expand \( \sin(n\pi + \theta) \) into \( \cos(n \pi) \sin \theta + \sin(n \pi) \cos \theta \).

Step 2: Apply sine and cosine of multiples of pi

Evaluate each term separately. For any integer \( n \), \( \cos(n \pi) \) is \( (-1)^n \) and \( \sin(n \pi) \) is \( 0 \). Substituting these will result in the expression \( (-1)^n \sin \theta + 0 \cdot \cos \theta \).

Step 3: Simplify the expression

Simplify the expression from the step 2 to get \( (-1)^n \sin \theta \), which is exactly the right-hand side of the given equation, showing that the identity holds true.

Step 4: Recap the results

Going back to the original identity \( \sin(n\pi + \theta) = \left(-1\right)^n \sin \theta \), we find that both sides are equal, thus the identity is verified.

Key Concepts

Understanding the Sine Function

The sine function, denoted as \( \sin \theta \), is one of the fundamental trigonometric functions. It describes the ratio of the opposite side to the hypotenuse in a right triangle. It's also crucial when dealing with circular and harmonic motion.
In any unit circle, \( \sin \theta \) represents the y-coordinate of a point. This is especially useful in transformations and oscillatory problems.

• The sine curve is periodic with a period of \( 2\pi \).
• Its range is between -1 and 1.
• Key points to remember: at \( \theta = 0 \), \( \sin \theta = 0 \); at \( \theta = \pi/2 \), \( \sin \theta = 1 \).

Understanding sine is pivotal for solving problems in both basic trigonometry and advanced calculus.

Exploring the Cosine Function

The cosine function, represented as \( \cos \theta \), is another essential trigonometric function. It expresses the ratio of the adjacent side to the hypotenuse in a right triangle. Like sine, it helps describe wave and oscillatory behavior.
In the unit circle, \( \cos \theta \) corresponds to the x-coordinate. Together with sine, cosine forms the foundation of harmonic analysis.

• The cosine curve is also periodic, with a period of \( 2\pi \). However, it starts at 1 when \( \theta = 0 \).
• Its values range from -1 to 1.
  
• Some important angles include \( \cos 0 = 1 \) and \( \cos \pi = -1 \).

The cosine function's symmetry and periodic nature make it indispensable in trigonometry.

Integer Multiples of Pi and Their Role

When dealing with integer multiples of \( \pi \), both sine and cosine functions display unique values.
Specifically, \( \cos(n \pi) \) results in \( (-1)^n \), and \( \sin(n \pi) \) equals 0, where \( n \) is an integer. These properties are integral in simplifying and verifying trigonometric identities.

• For even \( n \), \( \cos(n \pi) = 1 \).
• For odd \( n \), \( \cos(n \pi) = -1 \).
• For any integer \( n \), \( \sin(n \pi) = 0 \).

Understanding these concepts makes it easier to solve and manipulate trigonometric expressions like those in this exercise.

The Sum of Angles Identity in Trigonometry

The sum of angles identity is a valuable tool in trigonometry. It allows breaking down sine and cosine of summed angles into simpler parts.
For sine, the identity is:\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
This identity is crucial for expanding and simplifying expressions where angles are added.

• Helps in transforming complex expressions into understandable parts.
• Used in scenarios like wave interference and resonances.

Applying this identity helped in the solution to simplify \( \sin(n\pi + \theta) \) to match the given trigonometric identity.