Problem 33 - Precalculus with Calculus Previews

Question

Justify the given statement with one of the properties of the trigonometric functions. $$ \sin \pi=\sin 3 \pi $$

Step-by-Step Solution

Answer

\(\sin \pi = \sin 3\pi\) due to the periodicity of sine function.

Step 1: Understand the Trigonometric Identity

The statement states that \(\sin \pi = \sin 3\pi\). To verify this, recall that sine is a periodic function with period \(2\pi\). This means that for any angle \(\theta\), \(\sin(\theta) = \sin(\theta + 2k\pi)\), where \(k\) is any integer.

Step 2: Apply the Periodicity Property

Using the property from Step 1, calculate \(\sin 3\pi\): \(3\pi = \pi + 2\pi\), so we can rewrite this as \(\sin 3\pi = \sin(\pi + 2\pi)\). By the periodicity property, this simplifies to \(\sin \pi\).

Step 3: Conclusion

Since both \(\sin \pi\) and \(\sin 3\pi\) simplify to the same value using the periodicity property, the statement \(\sin \pi = \sin 3\pi\) is justified. The sine function’s periodicity property confirms that these two expressions are equal.

Key Concepts

The Sine Function

The sine function is one of the fundamental trigonometric functions. It relates angles of a right triangle to the ratio of the length of the opposite side to the hypotenuse. This relationship makes it a crucial tool in geometry and physics. For any given angle \( \theta \), the sine function is denoted as \( \sin \theta \). The sine function is not limited to angles between 0 and 90 degrees. It can be used for any real number, as it is a periodic function, meaning it repeats its values in a regular pattern.
The graph of the sine function is a smooth, continuous wave that oscillates above and below the x-axis. This wave-like pattern continues indefinitely in both the positive and negative x-directions. The sine function has a range from -1 to 1, meaning the output value of the sine function will always lie within this interval.

• The function reaches its maximum value, 1, at angles that are odd multiples of \( \frac{\pi}{2} \) (such as \( \frac{\pi}{2}, \frac{5\pi}{2} \)).
• It reaches its minimum value, -1, at angles that are odd multiples of \( \frac{3\pi}{2} \) (such as \( \frac{3\pi}{2}, \frac{7\pi}{2} \)).
• It passes through zero at integer multiples of \( \pi \) (such as \( \pi, 2\pi, 3\pi \)).

Understanding the behavior of the sine function is essential for solving many trigonometric concepts and calculations, such as justifying identical values at different angles.

Periodicity Property

The periodicity property of the sine function is a crucial concept in trigonometry. Because the sine function oscillates in a wave-like manner, it repeats its values at regular intervals. This interval is called the period of the function. The sine function has a period of \( 2\pi \), meaning after every \( 2\pi \) radians, the function's values cycle back to the start of the pattern.
For instance, if you find that \( \sin \theta = \sin(\theta + 2k\pi) \), with \( k \) being any integer, you're observing the periodic nature of sine. This property allows us to simplify expressions involving angles that are larger than \( 2\pi \) or even negative by reducing them to an equivalent angle within a single cycle of the sine wave.
Consider \( \sin 3\pi \). Using the periodic property, you can express this angle as \( \pi + 2\pi \). Since \( \sin(\theta + 2\pi) = \sin \theta \), it follows that \( \sin 3\pi = \sin \pi \). Therefore, these two expressions are equal due to the periodic nature of the sine function.

Trigonometric Identity

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable involved. These identities are fundamental in simplifying trigonometric expressions and solving equations. Some common trigonometric identities include the Pythagorean identity, angle sum and difference identities, and double angle formulas.
In the context of the given exercise, one of the identities we leverage is related to periodicity. The statement \( \sin \pi = \sin 3\pi \) can be verified using the periodic nature of the sine function. By understanding that \( \sin \theta = \sin(\theta + 2k\pi) \) for any integer \( k \), we can deduce the equality of these two sine values. This is because \( 3\pi \) is simply \( \pi + 2\pi \), fitting neatly into the identity.

• These identities simplify complex trigonometric problems and aid in transforming expressions into more workable forms.
• Mastering trigonometric identities is essential for effective problem-solving in mathematics, physics, and engineering.

Understanding and applying these identities helps in confirming the equivalence of expressions, much like in verifying the periodic nature of the sine values in the exercise.