Simplify arctan(-1/3* square root of 3)
The given question is asking you to simplify the expression "arctan(-1/3 * √3)," which involves the inverse trigonometric function arctan, or arctangent. The argument of the function is a numeric fraction that involves the square root of 3. Simplification of this expression entails evaluating the arctangent of a specific negative rational value that has been multiplied by the square root of 3. The aim is to find the angle whose tangent is equal to this negative fraction, possibly reducing the expression to a known angle or a representation that indicates the angle in terms of standard positions on the unit circle.
$arctan \left(\right. - \frac{1}{3} \cdot \sqrt{3} \left.\right)$
Answer
Solution:
Step 1:
Merge the terms $\sqrt{3}$ and $\frac{1}{3}$ to form $arctan\left(-\frac{\sqrt{3}}{3}\right)$.
Step 2:
Determine the precise value for $arctan\left(-\frac{\sqrt{3}}{3}\right)$, which is $-\frac{\pi}{6}$.
Step 3:
Express the solution in various formats. In its exact form, the solution is $-\frac{\pi}{6}$. In decimal form, it is approximately $-0.52359877$.
Knowledge Notes:
The problem involves simplifying the inverse trigonometric function arctan, or arctangent, of a negative rational expression involving a square root. Here are the relevant knowledge points:
Inverse Trigonometric Functions: These functions, including arctan, are the inverses of the trigonometric functions. They return the angle whose trigonometric function equals the given number. For example, if $tan(\theta) = x$, then $arctan(x) = \theta$.
Simplifying Expressions: The process of combining terms and simplifying expressions is a fundamental algebraic skill. In this case, the expression inside the arctan function is simplified by combining the square root and the fraction.
Exact Values of Trigonometric Functions: Certain angles have trigonometric values that can be expressed exactly in terms of $\pi$. For example, $arctan\left(-\frac{\sqrt{3}}{3}\right)$ corresponds to the angle $-\frac{\pi}{6}$ because $tan\left(-\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{3}$.
Radians and Degrees: Trigonometric functions can be expressed in radians or degrees. In this solution, the angle is given in radians, where $\pi$ radians is equivalent to 180 degrees.
Decimal Approximations: Trigonometric functions often result in irrational numbers, which can be approximated in decimal form for practical use. However, it is important to remember that these are approximations and not exact values.
Understanding these concepts is crucial for solving problems involving inverse trigonometric functions and simplifying expressions within them.
