en angles are small, we often use the "small-angle approximation": we use simply e tin In...
The small angle approximation is often made to simplify derivations and calculations. For the following angles θ, compare the true value of the sine of the angle to the to the small angle approximation (using only the first term in the series expansion of the sine) by determining the fractional error (let the fractional error be positive). NOTE: If you explicitly use π to convert from degrees to radians, use an accurate value for it. DIGRESSION: The fractional error is...
When angles are small |0| < 10° we often make the small angle approximation that sin Orad ñ Orad, where Orad is the angle measured in radians. Often, this approximation allows us to determine stability, or to more easily interpret solutions for these small angles; for example, the small angle approximation is what allows us to approximate the simple pendulum as a simple harmonic oscillator. Compute the difference sin Orad – Orad for the following angles: a) Odeg = 0°,...
In case of angle x is small (x < 5o), we can use the following approximation to determine sine and tangent of x: sin(x)=tan(x)=x (in radian) Applying approximation above to calculate sine of 3o 0.052 0.053 0.0009 3.000
A low-loss dielectric is one with a small loss tangent, that is, (o/we) <1. We can reduce (6.52) for this special case by applying a binomial series expansion to the value within the 6.3.1 Low-Loss Dielectrics interior square root portion of the equations. The expansion is n(n-1);- + (1 + x)" = 1 + nx + 2! and for x< 1 this can be approximated as (1 + x)" = 1+ nx So we have + 1+ (6.53) 08 Inserting...