Definition: An angle of $1$ radian is defined to be the angle, in the counterclockwise direction, in ~ the facility of a unit circle i beg your pardon spans an arc of length $1.$ The picture below illustrates this definition.

You are watching: Find the corresponding angle measure in radians on the unit one below, the angles $10^\\circ,\\ 20^\\circ,\\ 30^\\circ,$ etc., are indicated by black color dots top top the circle. Mark off angle of measure up 0, 1, 2, 3, 4, 5, and 6 radians. Calculation the equivalent angle measure up in degrees. Estimate the angle in radians the correspond come $180^\\circ$ and $360^\\circ.$

## IM Commentary

Radians are often mysterious come students, yet they space a an extremely straight forward method to measure up an edge by relating the measure up of the edge to the size of the arc top top the unit circle that subtends. V this definition, the angle that space easily explained in radians have units that $\\pi$ - a full transformation of the circle has an angle of $2\\pi$ radians. This job is not designed to uncover the meaning of radian, fairly it allows students come make an interpretation out the the definition. The idea is that students use a item of string to measure up the radius the the circle and also then note off the corresponding arc length on the unit circle as plenty of times together they need to for 0, 1, 2, 3, 4, 5, and 6 radians. Castle then deserve to use the black color dots to estimate the corresponding angle measure up in degrees.

The job does not specifically cite the usage of string. Students can use a ruler or a item of file to \"bend the radius about the circle\". This offers an opportunity to use appropriate tools and also talk about precision (SMP 5 and also 6).

A various solution method takes benefit of transformations, specifically rotations. In this solution we reduced out the snapshot of a unit circle v one radian marked and put it on optimal of the unit one on the grid. When one radian is marked off ~ above the circle, we have the right to mark turn off 2, 3, 4,... Radians (approximately) through rotating the cut out one so that the old starting point of the 1 radian arc sit on optimal of the old finish point.

After students uncover the approximate radian measure up for $180^\\circ$ and also $360^\\circ$ the teacher can suggest out that the exact corresponding angles in radians space $\\pi$ and also $2\\pi,$ respectively.

## Solution

We deserve to use a item of cable or a bendable ruler to measure up the radius that the circle and trace the very same distance roughly the circumference of the circle because that 1, 2, 3, 4, 5, and 6 radians. The diagram listed below shows the matching angles. We have the right to see from the diagram the $180^\\circ$ synchronizes to a small bit much more than 3 radians. In fact, we deserve to use the 10 degree points to aid us obtain a better estimate. The arc subtended by 1 radian is separated into a small less than 6 equal pieces by the points and also the arc subtended through 3 radians is about 1 piece brief of half the circle, for this reason $180^\\circ$ is around $3+\\frac16 \\approx 3.17$ radians. Similarly, $360^\\circ$ is around 6.34 radians: these room both slight end estimates due to the fact that the arc subtended through 3 radians is a little bit much less than 1 piece quick of $180^\\circ$.

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Definition: An edge of $1$ radian is defined to be the angle, in the counterclockwise direction, in ~ the center of a unit circle which spans an arc of size $1.$ The snapshot below illustrates this definition. top top the unit circle below, the angle $10^\\circ,\\ 20^\\circ,\\ 30^\\circ,$ etc., are indicated by black color dots top top the circle. Note off angle of measure 0, 1, 2, 3, 4, 5, and 6 radians. Calculation the matching angle measure up in degrees. Estimate the angle in radians the correspond come $180^\\circ$ and also $360^\\circ.$