![]() If you liked that first exercise, you may want to try the power series for cos(x) with x = π/2 = 1.5708 and see how fast it converges to 0. Īs an exercise, you may want to take the power series for sin(x) where x is in radians, substitute x = π/6 = 0.5236 and see how close it comes to 0.5, the correct value for sin(30°) as you vary the number of terms. The same power series would be less "elegant" (a favorite word of mathematicians) if angles are measured in degrees: These are infinite series that converge to the sine and cosine functions. Why are radians more natural than degrees? Comparing the first and second above expressions for s(t), it is easier to see that the frequency is 3 kHz (3000 cycles per second) when angles are measured in radians – provided I leave the 2π factor as an explicit multiplier, which mathematicians and engineers always do.Ī second, more fundamental reason that radians are more natural can be seen from the "power series" for sine and cosine. That's because there are 360° in a cycle (a full circle) so 3000 cycles per second equalsģ000 cycles/sec x 360°/cycle = 1,080,000°/sec. Which is the equivalent expression when angles are measured in degrees. To express one of the signals in our discussion with angles measured in radians, I could instead have written You should get 360° to four significant figures.)ģ0°, which is 1/12 of a full circle, therefore equals 2π/12 = π/6 radians and sin(π/6) = 0.5ĩ0°, which is 1/4 of a full circle, therefore equals 2π/4 =π/2 radians, and cos(π/2) = 0 Check it out by multiplying 57.30° by 2π = 6.283. Since 90° = π / 2 radians, to four significant figures, one radian equals 180°/ π = 57.30°. Also, since an angle in radians is defined as the ratio of two lengths, L/r, it is dimensionless. We usually suppress the unit of measurement "radians" since it is understood if no other units for angles is specified. The circumference of the entire circle is (2 π r) the arc length of the 1/4 of that circle subtended by this angle is L = (2 π r) / 4 = (π r) / 2 and the ratio of that arc length L to the radius r is π / 2. To put that mouth full of words into a diagram, the figure below shows an angle of 90°, the arc length L subtended by that angle, and the radius r of the circle. B ut it turns out that a more natural measure for angles, at least in mathematics, is in radians.Īn angle measured in radians is the ratio of the arc length of a circle subtended by that angle, divided by the radius of the circle. ![]() For example, there are 360° in a full circle or one cycle of a sine wave, and sin(30°) = 0.5 and cos(90°) = 0. Most of the time we measure angles in degrees.
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