Problem 26 - Algebra and Trigonometry

Question

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 exact value is 1.

Step 1: Understand the Inverse Sine Expression

We need to find the angle \( \theta \) such that \( \sin(\theta) = \frac{\sqrt{2}}{2} \). We know from trigonometric values that \( \theta = \frac{\pi}{4} \) because \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \).

Step 2: Interpret the Expression

Since \( \theta = \frac{\pi}{4} \) when \( \sin^{-1} \left( \frac{\sqrt{2}}{2} \right) \), we can rewrite the original expression as \( \tan \left( \frac{\pi}{4} \right) \).

Step 3: Evaluate the Tangent Function

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

Key Concepts

Understanding the Tangent Function

The tangent function, often denoted as \( \tan(\theta) \), is a fundamental trigonometric function. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. More formally, given an angle \( \theta \) in the unit circle, \( \tan(\theta) \) can be expressed as:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

This identity is particularly useful as it signifies the relationship between the sine and cosine functions. The tangent function is periodic with a period of \( \pi \), meaning \( \tan(\theta + \pi) = \tan(\theta) \) for all \( \theta \). This periodic nature helps in understanding its behavior across different quadrants of the unit circle. A unique quality of the tangent function is its undefined nature at \( \theta = \frac{\pi}{2} + n\pi \) for integers \( n \), where \( \cos(\theta) \) equals zero. When calculating specific values, like \( \tan\left(\frac{\pi}{4}\right) \), it helps to know common angle values, where \( \tan\left(\frac{\pi}{4}\right) = 1 \). These values stem from the 45-45-90 triangle properties, signifying the equal length of the opposite and adjacent sides for \( \theta = \frac{\pi}{4} \).

Exploring the Inverse Sine

The inverse sine function, written as \( \sin^{-1}(y) \) or \( \arcsin(y) \), is the function that "undoes" sine. It's also known as "arcsine." Inverse sine finds an angle whose sine is \( y \), within the range of \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), which includes the first and fourth quadrants. For example, if \( \sin(\theta) = y \), then \( \sin^{-1}(y) = \theta \). It's crucial to note that the range is limited to ensure the function remains a valid inverse, making it single-valued and continuous. In this context, when asked to find \( \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) \), we recognize it as \( \frac{\pi}{4} \) since \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). The inverse sine, by its nature, will only provide this principal value, which is the most direct and positive angle when working with trigonometric equations.

Understanding Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the included variables. These identities are essential tools in solving trigonometric expressions and equations. One of the primary trigonometric identities is \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This identity helps in deriving other trigonometric values from known sine and cosine values.

• Pythagorean identities, like \( \sin^2(\theta) + \cos^2(\theta) = 1 \), often serve as the foundation for simplifying expressions and solving equations.
• Another useful identity is the angle sum identity, \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \), which allows calculating the tangent of angle sums from their individual values.

Utilizing trigonometric identities effectively allows simplification and transformation of equations, making complicated expressions into manageable forms. These identities also lead to insights about the connected nature of the trigonometric functions, enhancing a deeper understanding across their individual and collective behaviors.