Problem 43 - Algebra and Trigonometry

Question

Find the exact value of each expression, if possible. Do not use a calculator. $$ \sin ^{-1}(\sin \pi) $$

Step-by-Step Solution

Answer

The exact value of \( \sin^{-1}(\sin \pi) \) is 0.

Step 1: Compute the Sine of pi

Start by computing the sine of pi. Note that pi in radians corresponds to 180 degrees, and the sine of 180 degrees or pi radians is 0. So, \( \sin (\pi) = 0 \).

Step 2: Compute Inverse Sine

Now, substitute the value from Step 1 into the expression as input for the arcsine function. This gives \( \sin^{-1}(0) \). The arcsine of a value is the angle whose sine gives that value. Now, we need to find an angle whose sine is 0. This angle is 0 (and additionally pi, but since the range of arcsine is typically defined from -pi/2 to pi/2, we take 0). So, \( \sin^{-1}(0) = 0 \).

Step 3: Final Answer

Based on the operations in the previous steps, it has been determined that the final answer to \( \sin^{-1}(\sin \pi) \) is 0. This is because the sine of pi is 0, and the arcsine of 0 is 0.

Key Concepts

Trigonometric Identities

Trigonometric identities are relationships between the trigonometric functions that hold true for all angles. They are very useful for simplifying expressions, solving equations, and proving other mathematical statements. In this exercise, we used the identity that relates angles to their trigonometric functions. For instance, we know that

• The sine of an angle gives the vertical position of a point on the unit circle.
• The sine function creates a wave pattern, peaking at 1 and troughing at -1, which repeats every 2π radians.

These identities include some basic ones such as:

• Pythagorean identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Negative angle identities: \( \sin(-x) = -\sin(x) \)

In our problem, the sine of π is evaluated. By identity, \( \sin(\pi) = 0 \),showing how the wave returns to the axis every π.Understanding these identities can help to solve trigonometric problems without calculators. Keep these identities in mind as you work with trigonometric problems, and remember that they can often provide shortcuts or a more intuitive understanding of the relations between angles and their functions.
Every identity can be derived and applied to various scenarios encountered in mathematics and its applications.

Sine Function

The sine function is one of the core trigonometric functions and is essential in understanding periodic phenomena. It relates an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. The function is periodic and is usually expressed as

• \( y = \sin(x) \)
• It oscillates between -1 and 1, repeating every 2π radians.

This means that after every 2π interval, the sine function starts again from its original position. Considering specific values like \( \sin(0), \sin(\pi/2), \sin(\pi) \)which are 0, 1, and 0 respectively

• These evaluations show how the sine function traverses the wave starting from zero.

This is important when dealing with waves, vibrations, or circular motion, as it perfectly models these regular and repeating behaviors.
In the initial exercise, we utilized the sine of π, which is 0, demonstrating this periodic characteristic of the sine function. Remember that the sine function is not limited only to angles in triangles but applies broadly in contexts like circular motion and signal processing across different fields of study.

Inverse Functions

Inverse trigonometric functions are the reverse of the trigonometric functions. They take the result of a trigonometric function and return an angle. Arcsine, denoted \( \sin^{-1}(x) \),is particularly interesting. It finds the angle whose sine value is 'x'. Typically, the range of the arcsine is defined from \( -\frac{\pi}{2} \)to\( \frac{\pi}{2} \).Let's see what happens when we apply \( \sin^{-1} \)) to a value.

• For example, the arcsine of 0 is 0, since it refers to an angle whose sine is zero within the specified range.

The exercise involves calculating \( \sin^{-1}(\sin \pi) \).
Since \( \sin \pi \)is equal to 0, the problem becomes \( \sin^{-1}(0) \),which within the standard arcsine range gives us 0 as the answer.
Inverse functions are useful for finding angles from known function values. Knowing when and how to use inverse trigonometric functions is necessary for applications in calculus, physics, and engineering. They help in returning to the angle domain from the function's value domain, crucial for solving real-world problems requiring angle calculations.