Arctangent

Definition

The arctangent is the inverse trigonometric function that calculates the angle whose tangent is a given value. It is used to determine the angle in the standard position of a right triangle when the tangent ratio is known.

5 Must Know Facts For Your Next Test

1. The arctangent function is denoted as $\tan^{-1}(x)$ or $\arctan(x)$, where $x$ is the tangent ratio.
2. The arctangent function returns an angle in the range of $-\pi/2$ to $\pi/2$ radians, or $-90^\circ$ to $90^\circ$.
3. Arctangent can be used to solve trigonometric equations by isolating the arctangent term and applying the inverse function.
4. The domain of the arctangent function is all real numbers, and the range is $(-\pi/2, \pi/2)$.
5. Arctangent is a one-to-one function, meaning each output value corresponds to a unique input value.

Review Questions

• Explain how the arctangent function is used to solve trigonometric equations.
  • To solve trigonometric equations using arctangent, the key is to isolate the tangent term on one side of the equation. Once the tangent ratio is isolated, the arctangent function can be applied to both sides to determine the angle whose tangent is the given value. This allows you to find the specific angle that satisfies the original trigonometric equation.
• Describe the relationship between the domain and range of the arctangent function.
  • The domain of the arctangent function is all real numbers, meaning it can accept any value as input. The range, however, is limited to the interval $(-\pi/2, \pi/2)$ radians or $(-90^\circ, 90^\circ)$. This is because the arctangent function returns the angle whose tangent is the given input value, and the tangent function is only defined in the first and third quadrants of the unit circle. Therefore, the arctangent function can only output angles within this restricted range.
• Analyze how the properties of the arctangent function, such as its one-to-one nature and restricted range, impact the process of solving trigonometric equations.
  • The one-to-one property of the arctangent function means that each output value corresponds to a unique input value. This is important when solving trigonometric equations, as it ensures that the arctangent function will return a single, unambiguous angle that satisfies the equation. Additionally, the restricted range of the arctangent function, limited to $(-\pi/2, \pi/2)$ radians, means that the solutions to trigonometric equations will also be confined to this interval. This can simplify the process of solving for the angles, as the possible solutions are narrowed down to a specific quadrant of the unit circle.

Related terms

Trigonometric Functions: The basic trigonometric functions, including sine, cosine, and tangent, describe the ratios of the sides of a right triangle.

Inverse Trigonometric Functions: Inverse trigonometric functions, such as arctangent, allow you to determine the angle given a specific trigonometric ratio.

Standard Position: The standard position of a angle is when the initial side of the angle is on the positive x-axis and the terminal side is in the first quadrant of the coordinate plane.