Students should have a deal of practice in finding the trigonometric functions of angles expressed in radians. Since students are more familiar with degrees, it is often best to convert back to degrees. Measuring angles in radians enables us to write down quite simple formulas for the arc length of part of a circle and the area of a sector of a circle. It should be stressed again that, to use this formula, we require the angle to be in radians.
The basic steps for solving trigonometric equations, when the solution is required in radians rather than degrees, are unchanged. Indeed, it is sometimes best to find the solution s in degrees and convert to radians at the end of the problem. Next page - Content - Graphing the trigonometric functions. Content Radian measure The measurement of angles in degrees goes back to antiquity. It fits nicely into the Babylonian base number system, and divides well by 2, 3, 4, 6, 10, 12, 15, 30, 45, 90… you get the idea.
A degree is the amount I, an observer, need to tilt my head to see you, the mover. Me in middle of track. You ran around. Selfish, right? Much of physics and life! Instead of wondering how far we tilted our heads, consider how far the other person moved. Degrees measure angles by how far we tilted our heads. Radians measure angles by distance traveled. So we divide by radius to get a normalized angle:. Moving 1 radian unit is a perfectly normal distance to travel. Strictly speaking, radians are just a number like 1.
Now divide by the distance to the satellite and you get the orbital speed in radians per hour. This formula only works when x is in radians! Well, sine is fundamentally related to distance moved , not head-tilting. Now imagine a car with wheels of radius 2 meters also a monster. Wow -- the car was easier to figure out than the bus! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed.
All because radians speak in terms of the mover. The reverse is easy too. How fast are the wheels turning? Time for a beefier example. Calculus is about many things , and one is what happens when numbers get really big or really small.
When you make x small, like. Even stranger, what does it mean to multiply or divide by a degree? Can you have square or cubic degrees? Radians to the rescue! If you go an even smaller amount, from 0 to.
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