Problem 195 - Precalculus

Question

For the following exercises, find the exact value without the aid of a calculator. $$\sin ^{-1}(1)$$

Step-by-Step Solution

Answer

The exact value of \(\sin^{-1}(1)\) is \(\frac{\pi}{2}\).

Step 1: Understand the Problem

We are asked to find the exact value of the inverse sine function, \(\sin^{-1}(1)\), which means we need to determine the angle whose sine value is 1.

Step 2: Recall the Definition of Inverse Sine

The function \(\sin^{-1}(x)\) returns the angle \(\theta\) such that \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\) and \(\sin(\theta) = x\). So, we need to find an angle within this range for which the sine is 1.

Step 3: Determine the Angle

Within the range \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\), the angle where \(\sin(\theta) = 1\) is \(\theta = \frac{\pi}{2}\), because \(\sin\left(\frac{\pi}{2}\right) = 1\).

Key Concepts

Sine Function

The sine function, typically represented as \( \sin(\theta) \), is one of the fundamental trigonometric functions. It's used to relate the angle of a right triangle to the ratio of the lengths of specific sides of the triangle. In a right triangle:

• \( \sin(\theta) \) is equal to the length of the side opposite to angle \( \theta \) divided by the hypotenuse.
• This function oscillates between -1 and 1, as \( \theta \) ranges from \( 0 \) to \( 2\pi \) radians (or \( 0 \) to 360 degrees).

For example, if a picture of a right triangle is drawn, you would observe:

• The opposite side and hypotenuse changing as the angle changes.
• At \( \theta = 0\), the sine function starts at 0, peaks at 1 at \( \theta = \frac{\pi}{2} \), goes back to zero at \( \theta = \pi \), dips to -1 at \( \theta = \frac{3\pi}{2} \), and returns back to 0 at \( \theta = 2\pi \).

The periodic nature of sine is crucial, as it helps us understand wave behavior and oscillations.
This function is also pivotal in the study of calculus, physics, and engineering due to its ability to model repeated phenomena.

Exact Values

In trigonometry, certain angles are known to have specific 'exact values'. These are values that can be derived directly without using a calculator. Some commonly used exact values for sine and cosine functions are:

• \( \sin(0) = 0 \)
• \( \sin(\frac{\pi}{6}) = \frac{1}{2} \)
• \( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)
• \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \)
• \( \sin(\frac{\pi}{2}) = 1 \)

Knowing these values helps in solving trigonometric problems easily and efficiently. Angles that correspond to these values are often encountered in mathematics, especially when dealing with unit circles or certain geometric shapes. For instance, finding that \( \sin^{-1}(1) = \frac{\pi}{2} \) is simplified by knowing that the sine of \( \pi/2 \) is always 1. These values also help in estimating more complex angles and in understanding the symmetrical properties of trigonometric functions.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to the ratios of the sides of a right triangle. The main trigonometric functions include sine, cosine, and tangent, with their respective reciprocals being cosecant, secant, and cotangent. These functions are defined as:

• \( \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
• \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \)

Each function captures a unique ratio within a right triangle, which proves useful in many application areas such as physics, engineering, and signal processing. These functions are periodic, meaning they repeat their values in a predictable pattern as the angle \( \theta \) increases. This periodicity is used to model waves and circular motion. For example:

• The sine and cosine functions are both periodic with a period of \( 2\pi \).
• The tangent function has a period of \( \pi \).

Being familiar with the unit circle and the properties of these functions allows you to determine the values of these trigonometric functions for various angles quickly, aiding in tasks like graphing functions or solving equations.