Converting Radians and Degrees in Pre-Calculus

Converting Radians and Degrees in Pre-Calculus You are probably familiar with degrees as a measure of how large an angle is, but another way of describing angles is with radians. As you approach pre-calculus and your upper years of mathematics, degrees will become less and less frequent as radians become the norm, so it’s a good idea to get used to them early, especially if you plan on studying mathematics. Degrees work by dividing a circle into 360 equal parts, and radians work the same way, except a circle has 2Ï€ radians and  Ãâ‚¬Ã‚  or pi radians equal one-half of the circle or 180 degrees, which is important to remember. In order to convert angles from degrees to radians, then, students must learn to multiply the measurement of the degrees by pi divided by 180. In the example of 45 degrees in radians, one can simply reduce the equation of r 45Ï€ / 180 to  Ãâ‚¬/4, which is how you would leave the answer to express the value in radians. Conversely, if you know what an  angle is in radians and you want to know what the degrees would be, you multiply the angle by 180/Ï€, and thus 5Ï€ radians in degrees will equal 900 degrees- your calculator has a pi button, but in case its not handy, pi equals 3.14159265. Identifying Degrees and Radians Degrees are units of measurements valued one through 360 that measure the sections or angles of a circle while radians are used to measure the distance traveled by angles. Whereas there are 360 degrees in a circle, each radian of distance moved along the outside of the circle is equal to 57.3 degrees. Essentially, radians measure the distance traveled along the outside of the circle as opposed to the view of the angle that degree takes up, which simplifies solving problems that deal with measurements of distance traveled by circles like tire wheels. Degrees are much more useful for defining the interior angles of a circle than for how the circle moves or what distance is traveled by moving along the circle instead of merely looking at it from one perspective while radians are more appropriate for observing natural laws and applying to real-world equations. In either case, theyre both units of measurements which express the distance of a circle- its all a matter of perspective! The Benefit of Radians Over Degrees Whereas degrees can measure the internal perspective of angles of the circle, radians measure the actual distance of the circumference of a circle, providing a more accurate assessment of distance traveled than degrees which rely on a 360 scale. Additionally, in order to calculate the actual length of a segment of a circle with degrees, one must do more advanced computations that include the use of pi to arrive at a product. With radians, the conversion to distance is much easier because a radian views a circle from the perspective of distance rather than the measurement of internal angles alone. Basically, radians already factor in distance as part of the basis for the equation for defining a radians size, which makes them more versatile in use than degrees.

Spontaneous Process Definition and Examples

Spontaneous Process Definition and Examples In a system, whether it be in chemistry, biology, or physics there are spontaneous processes and nonspontaneous processes. Spontaneous Process Definition A spontaneous process is one that will occur without any energy input from the surroundings. It is a process that will occur on its own. For example, a ball will roll down an incline, water will flow downhill, ice will melt into water, radioisotopes will decay, and iron will rust. No intervention is required because these processes are thermodynamically favorable. In other words, the initial energy is higher than the final energy. Note how quickly a process occurs has no bearing on whether or not it is spontaneous. It may take a long time for rust to become obvious, yet when iron is exposed to air, the process will occur. A radioactive isotope may decay instantly or after thousands or millions or even billions of years. Spontaneous Versus Nonspontaneous Energy must be added in order for a nonspontaneous process to occur. The reverse of a spontaneous process is a nonspontaneous process. For example, rust doesnt convert back into iron on its own. A daughter isotope wont return to its parent state. Free Energy and Spontaneity The change in Gibbs free energy for a process may be used to determine its spontaneity. At constant temperature and pressure, the equation is: ΔG ΔH - TΔS Where ΔH is the change in enthalpy and ΔS is the change in entropy. If ΔG is negative, the process is spontaneous.If ΔG is positive, the process is nonspontaneous (but would be spontaneous in the reverse direction).If ΔG is 0 then the process is at equilibrium and no net change is occurring over time.