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Use local polynomial approximations for circular functions

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Small-angle approximations

The prescribed local relationships approximate circular functions near zero. The angle must be in radians, and the approximation is not treated as reliable far from zero

\[\sin x\approx x,\quad\cos x\approx1-\frac{x^2}{2},\quad\tan x\approx x\]
  1. 1Use radians Convert a degree measure to radians before applying a local relationship \(\displaystyle 5^{\circ}=5\times\frac{\pi\ \mathrm{rad}}{180}=\frac{\pi}{36}\ \mathrm{rad}\)
  2. 2Use the linear rule Near zero, sine and tangent are approximated by the radian argument \(\displaystyle \sin x\approx x,\quad\tan x\approx x\)
  3. 3Use the cosine rule Near zero, cosine is approximated by one minus half the square of the radian argument \(\displaystyle \cos x\approx1-\frac{x^2}{2}\)
  4. 4Check the local range Use the prescribed approximation inside the suitable local range and reject it in the clearly unsuitable range \(\displaystyle |x|\le0.1\ \mathrm{rad}\text{ is suitable},\quad |x|\ge0.5\ \mathrm{rad}\text{ is unsuitable}\)
Year 11, VCE Units 1 & 2, Chapter 9, 4 difficulty levels, Calculator allowedBuild a worksheet

Small-angle approximations

The prescribed local relationships approximate circular functions near zero. The angle must be in radians, and the approximation is not treated as reliable far from zero

\[\sin x\approx x,\quad\cos x\approx1-\frac{x^2}{2},\quad\tan x\approx x\]
  1. Use radians Convert a degree measure to radians before applying a local relationship \(\displaystyle 5^{\circ}=5\times\frac{\pi\ \mathrm{rad}}{180}=\frac{\pi}{36}\ \mathrm{rad}\)
  2. Use the linear rule Near zero, sine and tangent are approximated by the radian argument \(\displaystyle \sin x\approx x,\quad\tan x\approx x\)
  3. Use the cosine rule Near zero, cosine is approximated by one minus half the square of the radian argument \(\displaystyle \cos x\approx1-\frac{x^2}{2}\)
  4. Check the local range Use the prescribed approximation inside the suitable local range and reject it in the clearly unsuitable range \(\displaystyle |x|\le0.1\ \mathrm{rad}\text{ is suitable},\quad |x|\ge0.5\ \mathrm{rad}\text{ is unsuitable}\)

Worked example

Use the prescribed small-angle relationship to estimate the cosine value for the given radian argument

\[x=0.1\ \mathrm{rad},\quad\cos(x)\]
  1. \(\displaystyle x=0.1\ \mathrm{rad}\)The argument is already in radians and is within the suitable local range
  2. \(\displaystyle \cos x\approx1-\frac{x^2}{2}\)Use the prescribed quadratic approximation for cosine
  3. \(\displaystyle \frac{x^2}{2}=\frac{\left(0.1\right)^2}{2}=0.005\)Square the argument, then divide by 2.
  4. \(\displaystyle \cos(0.1)\approx1-0.005=0.995\)Substitute the calculated correction into the cosine approximation

Common mistakes

Using a linear cosine rule
✕\(\displaystyle \cos(x)\approx x\)✓\(\displaystyle \cos(x)\approx1-\frac{x^2}{2}\)
Cosine has a negative quadratic correction near zero, whereas sine and tangent use a linear approximation
Using degrees directly
✕\(\displaystyle \sin(5^{\circ})\approx5\)✓\(\displaystyle 5^{\circ}=\frac{\pi}{36}\ \mathrm{rad},\quad\sin(5^{\circ})\approx\frac{\pi}{36}\)
The prescribed relationships use the numerical angle in radians, not the degree number
Using the local rule outside its range
✕\(\displaystyle |0.6|\ge0.5\ \mathrm{rad},\quad\sin(0.6)\approx0.6\text{ is reliable}\)✓\(\displaystyle |0.6|\ge0.5\ \mathrm{rad}\Rightarrow\text{the local approximation is unsuitable}\)
The stated local approximation is unsuitable when the absolute radian argument is at least 0.5

Difficulty levels

introEvaluate sine, cosine or tangent at a small rational radian value
standardConvert a degree input to radians and compare an approximation at a stated accuracy
challengeSolve a simple equation using a local relationship or judge whether it is suitable
examSelect and apply an approximation, including a suitability decision

Questions about this topic

Why must the angle be in radians?+

These local relationships are derived using radian measure. A degree numeral such as 5 is not the same numerical angle as 5 radians

When is a small-angle approximation suitable?+

Under the prescribed rule, an absolute argument at most 0.1 radians is suitable and one at least 0.5 radians is unsuitable. The intervening values are not classified by this rule

Why does cosine have a quadratic term?+

Cosine starts at 1 and has zero first-order change at zero, so its leading local correction is negative and quadratic

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 9: Circular functions

In this chapter

Convert angle measure and use the radian arc relationLocate extended angles and generate coterminal anglesDeduce exact trigonometric ratios from one ratioUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsLocate and bisect a circular-equation rootUse local polynomial approximations for circular functions