Small Angle Approximations (for physics derivations and numerical problems)

For very small angles, you can approximate values for the sine, cosine, or tangent. This is useful when calculating the fringe separations in interference patterns.

When θ is measured in radians, a small angle segment approximates to s/r where s is the arc length and r is the radius. In other words, for a small angle segment, we can write, θ = s/r.

The rules for Small Angle Approximations are:

sin θ ≈ tan θ ≈ θ
cos θ ≈ 1

When θ is measured in radians, θ = s/r. To go round the circle once, the arc is 2πr, so the angle in radians is 2πr/r = 2π.
When θ is measured in radians, θ = s/r. [Small Angle Approximations for physics derivations and numerical problems]

To convert the angle in radians into an angle in degrees, remember that one radian is 180/π ≈ 57.3 degrees.

Exercise:

Use the small-angle rule to write down these values:
a) tan 0.01 radians
b) cos 0.05 radians
c) sin 0.03 radians.

Small Angle Approximations (for physics derivations and numerical problems)
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