Problem 23 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \sin \left(\operatorname{Arctan} \frac{\sqrt{3}}{3}\right) $$

Step-by-Step Solution

Answer

The value is \( \frac{1}{2} \).

Step 1: Identify the arctan value

The expression involves finding \( \sin(\operatorname{Arctan} \frac{\sqrt{3}}{3}) \). Start by identifying the angle \( \theta \) such that \( \tan(\theta) = \frac{\sqrt{3}}{3} \).

Step 2: Determine the reference angle

Recall that \( \tan(\theta) = \frac{1}{\sqrt{3}} \). From trigonometric identities, \( \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} \). Thus, the reference angle \( \theta = \frac{\pi}{6} \) radians.

Step 3: Find the sine of the reference angle

Now determine \( \sin(\frac{\pi}{6}) \). The sine of \( \frac{\pi}{6} \) is \( \frac{1}{2} \).

Step 4: Conclude the solution

Since \( \theta = \frac{\pi}{6} \) is correct, you have \( \sin(\operatorname{Arctan} \frac{\sqrt{3}}{3}) = \sin(\frac{\pi}{6}) = \frac{1}{2} \).

Key Concepts

Arctan

Understanding the concept of \(\text{arctan}\)is essential in trigonometry.This function, also known as the inverse tangent function, is written as\( \operatorname{Arctan}(x) \).It helps you find an angle\( \theta \)when the tangent of that angle is known.For example, if\( \tan(\theta) = \frac{\sqrt{3}}{3} \),we use\( \operatorname{Arctan} \)to find the angle\( \theta \).
\(\operatorname{Arctan}(x) \)will yield angles in radians between\(-\frac{\pi}{2} \)and\( \frac{\pi}{2} \).To solve an expression like\( \sin(\operatorname{Arctan}\frac{\sqrt{3}}{3}) \),it's necessary to find out what angle yields\( \tan\theta = \frac{\sqrt{3}}{3} \). This knowledge is crucial for tackling problems involving inverse trigonometric functions and simplifying expressions without guesswork.

Sine Function

The sine function is a fundamental part of trigonometry.It shows the relationship between the angle of a right triangle and the lengths of the triangle's sides.
In essence,the sine of an angle\( \theta \)is the ratio of the length of the side opposite the angle to the length of the hypotenuse.Mathematically, it's written as\( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
For angles like\( \frac{\pi}{6} \)in radians,the sine is a common fraction, uncomplicated by square roots or complex numbers.Here,\( \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \).
This value means that for the angle\( \frac{\pi}{6} \),the opposite side is half the length of the hypotenuse.This simplicity and reliability make the sine function a go-to in calculations involving angles and geometry.

Trigonometric Identities

Trigonometric identities are vital tools in simplifying expressions and solving equations.They are equations involving trigonometric functions that are true for every value of the variable where both sides are defined.
For instance,knowing that\( \tan(\frac{\pi}{6}) = \frac{1}{\sqrt{3}} \)lets you identify related angles easily.These identities aren't just algebraic measurements.They are crucial for understanding how different angles compare across different trigonometric functions.
Here are some key identities:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)

Having these identities in your toolkit enables you to verify any unknown values, confidently navigate through problems, and engage with complex trigonometric problems with ease.

Angle in Radians

Angles can be measured in degrees or radians,but radians often make calculations easier in calculus and higher mathematics.
One full circle encompasses an angle of\( 2\pi \)radians, which is equivalent to 360 degrees.
This conversion - knowing that\( \pi \)represents 180 degrees -is integral in many mathematical contexts, including physics and engineering.

• For example, an angle of\( \frac{\pi}{6} \)is equivalent to 30 degrees.
• Similarly,\( \frac{\pi}{4} \)translates to 45 degrees.

In this exercise,expressing angles in radiansensures clarity and precision.It allows us to work seamlessly across various mathematical disciplines and apply these calculations in real-world contexts more efficiently.