Problem 30 - Precalculus : Building Concepts and Connections

Question

Evaluate each trigonometric function without the use of a calculator. $$\arcsin (\sin (5 \pi))$$

Step-by-Step Solution

Answer

The value of \( \arcsin (\sin(5 \pi))\) is 0.

Step 1: Identifying the repeating nature of the Sine function

The sine function has a period of \(2\pi\). This means it repeats every \(2\pi\). As a result, the sine of an angle and that of the angle plus any multiple of \(2\pi\) would be same. Hence,\( \sin(5\pi) = \sin(5\pi - 2 * 2\pi)\).

Step 2: Simplifying the expression

\( \sin(5\pi) = \sin(\pi)\)

Step 3: Applying the arcsine function

Now applying the arcsine function on both sides will enable us to compute the desired trigonometric function i.e \( \arcsin (\sin(5 \pi)) = \arcsin(\sin(\pi))\)

Step 4: Finding the arcsine of an angle

As \( \sin(\pi) = 0\), this leaves us with \( \arcsin(\sin(\pi)) = \arcsin(0)\) which is equal to 0.

Key Concepts

Sine Function

The sine function, often denoted as \( \sin \), is one of the six main trigonometric functions, which also include cosine, tangent, cosecant, secant, and cotangent. The sine function relates an angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. For any angle \( \theta \), the sine function is expressed as \( \sin(\theta) \).
In the context of circles, specifically the unit circle, the sine function represents the y-coordinate of a point on the circle.

• Periodic Nature: The sine function is periodic with a period of \( 2\pi \). This means that the values of the sine function repeat every \( 2\pi \) radians. Thus, \( \sin(\theta) = \sin(\theta + 2k\pi) \) for any integer \( k \).
• Key Values: Some commonly used values for the sine function include \( \sin(0) = 0 \), \( \sin\left(\frac{\pi}{2}\right) = 1 \), and \( \sin(\pi) = 0 \).

This periodicity can be particularly useful for simplifying expressions involving sine, such as finding \( \sin(5\pi) \), which can be simplified by considering the periodicity.

Arcsine Function

The arcsine function, denoted as \( \arcsin \) or \( \sin^{-1} \), is the inverse of the sine function. It enables you to determine the angle whose sine is a specific value. Due to its inverse nature, the arcsine function is crucial for solving equations where you need to find the angle corresponding to a given sine value.
Let’s explore some important aspects of the arcsine function:

• Domain and Range: The domain of the arcsine function is between -1 and 1, inclusive. This is because the sine function only produces values in this range. The range of \( \arcsin(x) \) is \([-\frac{\pi}{2}, \frac{\pi}{2}]\), meaning it will return angles within this interval.
• Unique Output: Arcsine is designed to give a unique angle, so when you find \( \arcsin(\sin(\theta)) \), it may not return the original angle \( \theta \) unless \( \theta \) already lies within its range \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Understanding the arcsine function is key for solving problems which involve reversing the process of finding a sine value to obtain the corresponding angle, as exemplified in the exercise with \( \arcsin(\sin(5\pi)) \).

Periodicity

Periodicity is a fundamental concept in trigonometry, referring to the repetitive nature of trigonometric functions over regular intervals. The most common periodic functions in trigonometry are sine and cosine, both of which have a period of \( 2\pi \).
Understanding periodicity helps in simplifying trigonometric expressions. For any trigonometric function \( f \), periodicity can be defined as:

• \( f(\theta + T) = f(\theta) \)
• where \( T \) is the period of the function, the smallest positive number for which this property holds.

In terms of sine, this periodicity means that:

• \( \sin(\theta + 2k\pi) = \sin(\theta) \) for any integer \( k \).

This characteristic allows us to adjust angles by adding or subtracting multiples of \( 2\pi \) without changing the sine value. It is particularly helpful in mathematical problems, such as those involving angles greater than \( 2\pi \) or less than \( 0 \), as seen in the evaluation of \( \sin(5\pi) \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the identity are defined. These identities are essential tools for simplifying trigonometric expressions and solving trigonometric equations.
Here are some key trigonometric identities:

• Pythagorean Identity: One of the most famous is \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Angle Sum and Difference Identities: For instance, \( \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta) \).
• Double Angle Formulas: Such as \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \).

These identities simplify complex trigonometric problems and reveal deep relationships between the functions, applicable in varied scenarios from geometry to calculus.
In our exercise, the understanding of periodicity and the arcsine function as inverses of sine demonstrates how identities can guide the handling of expressions like \( \arcsin(\sin(5\pi)) \).