Arcsec

Definition

Arcsec, short for arcsecond, is a unit of angular measurement that represents one-sixtieth of an arcminute, or one three-hundred-and-sixtieth of a degree. It is commonly used in fields such as astronomy, navigation, and surveying to express very small angles with high precision.

5 Must Know Facts For Your Next Test

1. Arcsec is commonly used in astronomy to measure the apparent size of celestial objects, such as stars and galaxies.
2. In trigonometric substitution, arcsec is often used as the substitution variable when the original variable is of the form $\sqrt{a^2 - x^2}$.
3. The inverse trigonometric function $\arcsin(x)$ can be expressed in terms of arcsec as $\arcsin(x) = \frac{\pi}{2} - \arcsec(\sqrt{1 - x^2})$.
4. Arcsec is a useful unit for measuring small angles, as it allows for more precise measurements compared to degrees or radians.
5. The conversion between arcsec and degrees is 1 degree = 3600 arcsec, and the conversion between arcsec and radians is 1 radian = 206,265 arcsec.

Review Questions

• Explain how arcsec is used in the context of trigonometric substitution.
  • In the context of trigonometric substitution, arcsec is often used as the substitution variable when the original variable is of the form $\sqrt{a^2 - x^2}$. This is because the trigonometric identity $\sec(x) = \frac{1}{\cos(x)}$ can be used to express $\sqrt{a^2 - x^2}$ in terms of the secant function, which is the reciprocal of the cosine function. By making the substitution $x = a\sec(u)$, the original integral can be simplified and more easily integrated.
• Describe the relationship between arcsec and the inverse trigonometric function $\arcsin(x)$.
  • The inverse trigonometric function $\arcsin(x)$ can be expressed in terms of arcsec as $\arcsin(x) = \frac{\pi}{2} - \arcsec(\sqrt{1 - x^2})$. This relationship is useful in simplifying integrals involving the inverse sine function, as the arcsec function can be more easily integrated. By making the substitution $x = \sin(u)$, the original integral can be transformed into one involving the arcsec function, which can then be integrated using standard techniques.
• Analyze the importance of arcsec in fields such as astronomy, navigation, and surveying.
  • Arcsec is a crucial unit of measurement in fields like astronomy, navigation, and surveying due to its ability to express very small angles with high precision. In astronomy, arcsec is used to measure the apparent size of celestial objects, such as stars and galaxies, which are often incredibly small from our perspective on Earth. In navigation and surveying, arcsec is used to accurately measure angles for tasks like mapping, land surveying, and celestial navigation. The high level of precision afforded by arcsec makes it an indispensable tool in these fields, where even the smallest angular measurements can have significant implications.

Related terms

Radian: A unit of angular measurement defined as the angle subtended by an arc equal in length to the radius of the circle. One radian is approximately equal to 57.3 degrees.

Arcminute: A unit of angular measurement equal to one-sixtieth of a degree, or 60 arcseconds.

Trigonometric Substitution: A technique used in calculus to simplify integrals by replacing the original variable with a trigonometric function, allowing for easier integration.