Small Angle Approximations
Near zero, and only in radians, the sine of an angle is the angle, the tangent of it is the angle too, and the cosine drops off as one minus half the square. These are the approximations that turn an equation nobody can solve into one anybody can, and knowing when they stop being safe is half the point.
What a learner can do afterwards
- States the three approximations, insists on radians, and says what goes wrong in degrees
- Reads the approximations off the graphs of sine, cosine and tangent near the origin
- Uses an approximation to simplify an expression, then checks the size of the error it introduced at a stated angle
1 · Read
Near zero, and only in radians, three shortcuts hold: sin x behaves like x, tan x behaves like x too, and cos x behaves like 1 minus x squared over 2. Radians are non negotiable because arc length equals radius times angle only in radians. Degree based slopes would drag pi over 180 factors into every derivative, so convert to radians first and approximate second.
The graphs show why the rules work. Sine hugs the diagonal y equals x near the origin, since both share the same value and the same slope there. Cosine peaks flat at 1 and falls off quadratically, so 1 minus cos x behaves like x squared over 2. Since sin behaves like x and cos behaves like 1, their ratio tan behaves like x.
Estimate cos 0.2 as 1 minus 0.04 over 2, which is 0.98. Estimate cos 0.1 as 1 minus 0.01 over 2, which is 0.995. Replace sin 0.06 with 0.06 and the fraction sin 0.06 over 0.06 comes out about 1. After replacing sin 0.3 with 0.3, the next Taylor term x cubed over 6 prices the mistake at 0.0045.
Always quote the value with its price. The dropped terms measure the damage: x cubed over 6 bounds sine error and x to the fourth over 24 bounds cosine error. Halving x shrinks the sine error bound eightfold, so smaller angles are dramatically safer.
In radians, sine and tangent mimic x while cosine dips quadratically, each with a priced error.
More to read
6.3 Taylor and Maclaurin Series
Taylor series are polynomials tailored to hug a function near one point. For sine the tailor keeps only odd powers: x - x cubed/6 and onward, which is why sin x behaves like x up close. For cosine only even powers survive: 1 - x squared/2 onward, which is why cos x behaves like 1 minus a small quadratic dip. Truncating after the first surviving term is the whole approximation game. The dropped terms measure the damage. Alternating series bounds cap the error by the first neglected term, so x cubed/6 bounds sine error and x^4/24 bounds cosine error. Small x makes these caps tiny fast: halving x shrinks the sine error bound eightfold. Approximation plus error cap is the complete answer.
6.4 Working with Taylor Series
New series come from old by substitution and surgery. Replacing x with -x squared in the exponential series builds cosine style even series; dividing sine series by x term by term exposes the limit sin x/x = 1. These moves justify each small angle rule without fresh differentiation. One master series plus algebra generates the family. Error checks keep surgery honest. After replacing sin 0.3 with 0.3, the x cubed/6 term prices the mistake at 0.0045. Physics uses this pricing constantly: pendulums swing isochronously only while the cubic correction stays negligible. Quote the value and its price together.
4.2 Linear Approximations and Differentials
Linearisation is zooming until the curve looks straight. At the origin sine has slope 1, so its zoomed view is the diagonal y = x. Cosine has slope 0 there, so its linear view is the flat line y = 1 and the quadratic term carries the visible droop. Tangents give the linear rules; Taylor adds the curvature correction. Radians are non negotiable because arc length equals radius times angle only in radians. Degree based slopes would drag pi/180 factors into every derivative. Convert to radians first, approximate second, and check the error size last.
2 · Watch
En anglaisTake it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.