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Trigonometry

# Angle Measures

Angle measure in degrees: $$\alpha$$

Angle measure in radians: $$x$$

1. There are two commonly used units for measuring angles − degrees and radians. $$1$$ degree (denoted by $$1^\circ$$) is defined as$$1/360$$ of a complete revolution. The straight angle is equal to$$180^\circ$$, the right angle is $$90^\circ$$. The radian measure of an angle whose vertex lies at the centre of a circle is the ratio of the arc length to the radius of the circle. A central angle is equal to $$1$$ radian (denoted as $$1 \text{ rad }$$) if the angle subtends an arc whose length is equal to the radius of the circle.
2. $$1$$ degree contains $$60$$ minutes of arc: $$1^\circ = 60’$$. In turn, $$1$$ arcminute has $$60$$ arcseconds: $$1′ = 60^{\prime\prime}$$.
3. Value of $$1$$ radian in degrees
$$1 \text{ rad } = 180^\circ/\pi\ \approx$$ $$57^\circ 17’45^{\prime\prime}$$
4. Value of $$1$$ degree in radians
$$1^\circ = \pi/180 \text{ rad } \approx$$ $$0.017453 \text{ rad }$$
5. Value of $$1$$ arcminute in radians
$$1′ = \pi /\left( {180 \cdot 60} \right) \text{ rad } \approx$$ $$0.000291 \text{ rad }$$
6. Value of $$1$$ arcsecond in radians
$$1^{\prime\prime} = \pi /\left( {180 \cdot 3600} \right) \text{ rad } \approx$$ $$0.000005 \text{ rad }$$
7. Degrees to radians conversion $$x = \pi\alpha/{180^\circ},$$
where $$x$$ is the angle value in radians, $$\alpha$$ is the angle value in degrees.
8. Radians to degrees conversion $$\alpha = 180^\circ x/\pi,$$
where $$\alpha$$ is the angle value in degrees, $$x$$ is the angle value in radians.
9. Radian measures of common angles