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A-level Mathematics/CIE/Pure Mathematics 1/Circular Measure

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Radians

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Defining the radian

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The radian is defined as an angle subtended by an arc the same length as the radius.

A radian is a unit for measuring angles. It is defined as the angle subtended by an arc that is as long as the radius. As a consequence of this, there are radians in a full circle, because the length of the circumference is times the length of the radius.

Converting between radians and degrees

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Degrees are another common unit for measuring angles. There are in a circle, thus .

To convert from degrees to radians, multiply the number of degrees by .

e.g. is equal to

To convert from radians to degrees, multiply the amount of radians by .

e.g. radians is equal to

Geometric Calculations

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Arc length

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The length L is the arc. The area in green is the sector.

Arc length is, unsurprisingly, the length of a circular arc. This length depends on the size of the radius and the angle that the arc subtends.

We can think of an arc as a fraction of the circumference . This means that the arc length is the angle divided by a full circle times the length of the circumference: which can be simplified to .

e.g. The arc length for an arc with radius and angle is .

Sector areas

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The area of a sector can be derived in a similar way: it is a fraction of the area of a circle. The area of a circle is , so the area of a sector is .

Coordinate Geometry · Trigonometry