The length of the arc subtended by the central angle becomes the radian measure of the angle. This keeps all the important numbers like the sine and cosine of the central angle, on the same scale.
When graphing using degrees, the vertical scale must be stretched a lot to even see that the graph goes up and down.
But the utility of radian measure is even more obvious in calculus. To develop the derivative of the sine function you first work with this inequality At the request of a reader I have added an explanation of this inequality at the end of the post :. From this inequality you determine that. The middle term of the inequality is the area of a sector of a unit circle with central angles of radians. This limit is used to find the derivative of the sin x. Thus, with x in degrees,.
This means that with the derivative or antiderivative of any trigonometric function that is there getting in the way. Revision December 7, The inequality above is derived this way.
Consider the unit circle shown below. The central angle is and the coordinates of A are. Then the area of triangle OAB is. The area of sector. By similar triangles. Then the area of This is larger than the area of the sector, which establishes the inequality above. Multiply the inequality by and take the reciprocal to obtain. Finally, take the limit of these expression as and the limit is established by the squeeze theorem.
Thank you so much for this article. In Surveying, we use the Whole Circle Bearing WCB or Polar Coordinates, for all of our traverse calculations and convert that to rectangular coordinates to get the precise location of a point. But, I seem to keep falling short of the exact way in which the two work together to create the rectangular coordinate. Using the following sample numbers: Distance: Taking just the portion of the equations: COS Like Like.
Drop a perpendicular from the point to the x-axis. This forms a right triangle with hypotenuse of r. By right triangle trigonometry the length of this segment is and the distance from the pole to the foot of the perpendicular is. These same equations work for points in other quadrants and for angles with a negative measure i. Calculations may be done in degrees or radians as long as the calculator or computer you are using is set in the proper mode.
If your calculator or computer is set for degrees, there is no need to convert to radians. If your angles are measured in degrees, use degree mode on your device. Add comment. There is no such requirement Angle measure is arbitrary. The use of radians allows a direct relation between angular measure and linear measure. Ask a question for free Get a free answer to a quick problem.
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