# Convert arc second to hour angle

Learn how to convert 1 arc second to hour angle step by step.

## Calculation Breakdown

Set up the equation
$$1.0\left(arc \text{ } second\right)={\color{rgb(20,165,174)} x}\left(hour \text{ } angle\right)$$
Define the base values of the selected units in relation to the SI unit $$\left(radian\right)$$
$$\text{Left side: 1.0 } \left(arc \text{ } second\right) = {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}\left(radian\right)} = {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}\left(rad\right)}$$
$$\text{Right side: 1.0 } \left(hour \text{ } angle\right) = {\color{rgb(125,164,120)} \dfrac{π}{12.0}\left(radian\right)} = {\color{rgb(125,164,120)} \dfrac{π}{12.0}\left(rad\right)}$$
Insert known values into the conversion equation to determine $${\color{rgb(20,165,174)} x}$$
$$1.0\left(arc \text{ } second\right)={\color{rgb(20,165,174)} x}\left(hour \text{ } angle\right)$$
$$\text{Insert known values } =>$$
$$1.0 \times {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}} \times {\color{rgb(89,182,91)} \left(radian\right)} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} {\color{rgb(125,164,120)} \dfrac{π}{12.0}}} \times {\color{rgb(125,164,120)} \left(radian\right)}$$
$$\text{Or}$$
$$1.0 \cdot {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}} \cdot {\color{rgb(89,182,91)} \left(rad\right)} = {\color{rgb(20,165,174)} x} \cdot {\color{rgb(125,164,120)} \dfrac{π}{12.0}} \cdot {\color{rgb(125,164,120)} \left(rad\right)}$$
$$\text{Cancel SI units}$$
$$1.0 \times {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}} \cdot {\color{rgb(89,182,91)} \cancel{\left(rad\right)}} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} \dfrac{π}{12.0}} \times {\color{rgb(125,164,120)} \cancel{\left(rad\right)}}$$
$$\text{Conversion Equation}$$
$$\dfrac{π}{6.48 \times 10^{5}} = {\color{rgb(20,165,174)} x} \times \dfrac{π}{12.0}$$
Cancel factors on both sides
$$\text{Cancel factors}$$
$$\dfrac{{\color{rgb(255,204,153)} \cancel{π}}}{6.48 \times 10^{5}} = {\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{π}}}{12.0}$$
Switch sides
$${\color{rgb(20,165,174)} x} \times \dfrac{1.0}{12.0} = \dfrac{1.0}{6.48 \times 10^{5}}$$
Isolate $${\color{rgb(20,165,174)} x}$$
Multiply both sides by $$\left(\dfrac{12.0}{1.0}\right)$$
$${\color{rgb(20,165,174)} x} \times \dfrac{1.0}{12.0} \times \dfrac{12.0}{1.0} = \dfrac{1.0}{6.48 \times 10^{5}} \times \dfrac{12.0}{1.0}$$
$$\text{Cancel}$$
$${\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{1.0}} \times {\color{rgb(99,194,222)} \cancel{12.0}}}{{\color{rgb(99,194,222)} \cancel{12.0}} \times {\color{rgb(255,204,153)} \cancel{1.0}}} = \dfrac{{\color{rgb(255,204,153)} \cancel{1.0}} \times 12.0}{6.48 \times 10^{5} \times {\color{rgb(255,204,153)} \cancel{1.0}}}$$
$$\text{Simplify}$$
$${\color{rgb(20,165,174)} x} = \dfrac{12.0}{6.48 \times 10^{5}}$$
Rewrite equation
$$\dfrac{1.0}{10^{5}}\text{ can be rewritten to }10^{-5}$$
$$\text{Rewrite}$$
$${\color{rgb(20,165,174)} x} = \dfrac{10^{-5} \times 12.0}{6.48}$$
Solve $${\color{rgb(20,165,174)} x}$$
$${\color{rgb(20,165,174)} x}\approx0.0000185185\approx1.8519 \times 10^{-5}$$
$$\text{Conversion Equation}$$
$$1.0\left(arc \text{ } second\right)\approx{\color{rgb(20,165,174)} 1.8519 \times 10^{-5}}\left(hour \text{ } angle\right)$$