Problem 42 - Algebra and Trigonometry

Question

\(23-44=\) Find the exact value of the expression, if it is defined. \(\tan \left(\sin ^{-1} \frac{\sqrt{2}}{2}\right)\)

Step-by-Step Solution

Answer

The value of the expression is 1.

Step 1: Understand the Inverse Sine Function

Recognize that \(\sin^{-1}\left( \frac{\sqrt{2}}{2} \right)\) is the angle whose sine value is \(\frac{\sqrt{2}}{2}\). This is a commonly known angle in the unit circle which corresponds to \(\frac{\pi}{4}\) or \(45^\circ\).

Step 2: Determine the Tangent of the Angle

Next, we need to find \(\tan\left( \frac{\pi}{4} \right)\). From trigonometric identities, we know that \(\tan(\frac{\pi}{4}) = 1\).

Step 3: Combine the Steps

Combine the results from Step 1 and Step 2 to find \(\tan\left(\sin^{-1} \left(\frac{\sqrt{2}}{2}\right)\right) = \tan\left(\frac{\pi}{4}\right) = 1\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \sin^{-1}, \cos^{-1}, \) and \( \tan^{-1} \), allow us to find the angle that corresponds to a particular trigonometric ratio. When you see \( \sin^{-1}(x) \), it means "the angle whose sine is \( x \)." These functions are incredibly useful for solving problems involving angles and measurements.
For example, in this problem, \( \sin^{-1}(\frac{\sqrt{2}}{2}) \) represents the angle whose sine value is \( \frac{\sqrt{2}}{2} \). This is a familiar angle from the unit circle, namely \( \frac{\pi}{4} \) radians, or \( 45^{\circ} \).
These inverse functions have specific ranges to ensure that they return a unique value (angle). For instance, \( \sin^{-1}(x) \) returns values in the range of \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This makes them a powerful tool for finding exact angle values based on trigonometric ratios.

Unit Circle

The unit circle is a foundational concept for understanding trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle helps us visualize angles and their corresponding sine, cosine, and tangent values.
In the unit circle, each angle correlates to a point on the circle, with coordinates \((\cos(\theta), \sin(\theta))\). For the angle \(\frac{\pi}{4}\), the point is \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\). This is because both the sine and cosine of \(\frac{\pi}{4}\) are \(\frac{\sqrt{2}}{2}\).

• The circle allows us to immediately see the relationship between different trigonometric functions.
• It is easy to determine common values like \(\tan(\theta)\) by using \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).

The unit circle not only aids in understanding inverse functions but also highlights trigonometric phenomena such as periodicity and symmetry.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined. One fundamental identity is \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), which is crucial in solving the problem given.

• Knowing \(\sin(\theta)\) and \(\cos(\theta)\), you can directly find \(\tan(\theta)\).
• This identity helps balance expressions and simplify calculations.

The identity \(\tan(\frac{\pi}{4}) = 1\) is derived from this as both \(\sin(\frac{\pi}{4})\) and \(\cos(\frac{\pi}{4})\) are \(\frac{\sqrt{2}}{2}\), making their ratio exactly 1.
Utilizing these identities aids in transforming complex trigonometric expressions into simpler, more manageable forms, helping solve exercises efficiently.