Problem 39 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The value is \( \frac{\sqrt{3}}{3} \).

Step 1: Understand the Expression

The expression we need to solve involves an inverse trigonometric function inside a trigonometric function: \( \tan \left( \sin^{-1} \frac{1}{2} \right) \). We need to first evaluate the inside part \( \sin^{-1} \frac{1}{2} \).

Step 2: Evaluate the Inverse Sine

The expression \( \sin^{-1} \frac{1}{2} \) asks for the angle \( \theta \) whose sine is \( \frac{1}{2} \). From trigonometric knowledge, we know that the sine of \( \frac{\pi}{6} \) (or 30 degrees) is \( \frac{1}{2} \). Thus, \( \sin^{-1} \frac{1}{2} = \frac{\pi}{6} \).

Step 3: Find the Tangent of the Angle

Substitute \( \frac{\pi}{6} \) into the tangent function: \( \tan \left( \frac{\pi}{6} \right) \). The tangent of \( \frac{\pi}{6} \) is known to be \( \frac{1}{\sqrt{3}} \), which can also be written as \( \frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Key Concepts

Tangent of Angles

Tangent, often abbreviated as "tan," is one of the primary trigonometric functions that relates the angles of a triangle to the lengths of its sides. In a right triangle, the tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is expressed as:
\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]This function is very useful in various real-world applications like physics, engineering, and even in calculating heights and distances.

When dealing with angles outside the context of a triangle, such as in the unit circle, the tangent function can be extended. For any angle \( \theta \), the tangent is the sine of \( \theta \) divided by the cosine of \( \theta \):

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
• It is periodic, with a period of \( \pi \), meaning \( \tan(\theta + \pi) = \tan(\theta) \)
• The tangent function is undefined at angles where \( \cos(\theta) = 0 \)

Understanding tangent is crucial for evaluating expressions involving angles, especially when inversely related through functions like sine or cosine.

Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1}(x) \) or "arcsin," is used to find the angle whose sine is a given number. This is particularly useful when you know the sine value but need to find the corresponding angle.

For example, if you have \( \sin^{-1}(\frac{1}{2}) \), you're looking for the angle \( \theta \) such that \( \sin(\theta) = \frac{1}{2} \). The angle that satisfies this in the range \([-\frac{\pi}{2}, \frac{\pi}{2}]\) radians is \( \frac{\pi}{6} \) (or 30 degrees).

Key properties of the inverse sine function include:

• Its output range (also called the principal value) is restricted to \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
• It is the inverse of the sine function, meaning \( \sin(\sin^{-1}(x)) = x \) for \(-1 \leq x \leq 1\)
• Helps determine angles from known functions, acting as a bridge in solving trigonometric problems.

Inverse trigonometric functions like inverse sine open up the possibilities for solving equations that require returning from a numerical sine value to its angle form.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved where both sides are defined. They are powerful tools in simplifying expressions and solving trigonometric equations.

One of the most commonly used identities is the Pythagorean identity:

• \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

Besides the Pythagorean identity, there are sum and difference identities, double angle identities, and half angle identities, each serving different purposes in complex computations.

In our original problem, knowing that the tangent of \( \tan(\frac{\pi}{6}) \) simplifies to \( \frac{\sqrt{3}}{3} \), we are leveraging a trigonometric identity that relates tangent to sine and cosine:

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

Trigonometric identities help in transforming and solving trigonometric expressions by providing a straightforward path through seemingly complex calculations. Understanding these identities, their derivations, and applications is key to mastering trigonometry.