# Small Angle Approximations

When the angle θ (in radians) is small we can use these **approximations **for Sine, Cosine and Tangent:

If we are very daring we can use **cos θ ≈ 1 **

Let's see some values! (Note: values are approximate)

### sin θ ≈ θ

θ (radians) | sin θ | θ − sin θ |
---|---|---|

0 | 0 | 0 |

0.01 | 0.0099998 | 0.0000002 |

0.1 | 0.0998 | 0.0002 |

0.2 | 0.1987 | 0.0013 |

0.5 | 0.4794 | 0.0206 |

1 | 0.8415 | 0.1585 |

Perfect at zero, really good at 0.01, good at 0.1, and can be useful up to 0.5 if you aren't fussy.

### cos θ ≈ 1

Can we simply use **1** to approximate cos θ?

θ (radians) | cos θ | 1 − cos θ |
---|---|---|

0 | 1 | 0 |

0.01 | 0.99995 | 0.00005 |

0.1 | 0.995 | 0.005 |

0.2 | 0.9801 | 0.0199 |

0.5 | 0.8776 | 0.1224 |

1 | 0.5403 | 0.4597 |

Well yes we can, but only for very small angles.

### cos θ ≈ 1 − \frac{θ^{2}}{2}

So let us try the better version of 1 − \frac{θ^{2}}{2} :

θ (radians) | cos θ | 1 − \frac{θ^{2}}{2} | (1−\frac{θ^{2}}{2}) − cos θ |
---|---|---|---|

0 | 1 | 1 | 0 |

0.01 | 0.9999500004 | 0.99995 | -0.0000000004 |

0.1 | 0.9950042 | 0.995 | -0.0000042 |

0.2 | 0.980067 | 0.98 | -0.000067 |

0.5 | 0.8776 | 0.875 | -0.0026 |

1 | 0.5403 | 0.5 | -0.0403 |

Wow, that is a big improvement!

### tan θ ≈ θ

θ (radians) | tan θ | tan θ − θ |
---|---|---|

0 | 0 | 0 |

0.01 | 0.0100003 | -0.0000003 |

0.1 | 0.1003 | -0.0003 |

0.2 | 0.2027 | -0.0027 |

0.5 | 0.5463 | -0.0463 |

1 | 1.5574 | -0.5574 |

Not too bad for small values, right?

## Taylor Series

Did you see the magical improvement for cos when we went from **1** to **1 − \frac{θ^{2}}{2}** ?

The secret is the Taylor Series expansion of cos:

**cos x** = 1 − \frac{x^{2}}{2!} + \frac{x^{4}}{4!} − ...

So ... we can use more terms if want more accuracy!

Likewise we can improve sine:

**sin x** = x − \frac{x^{3}}{3!} + \frac{x^{5}}{5!} − ...

Or **tan**, or other functions like **e**^{x}

### Example: you are stuck on an island without a calculator. Calculate sine of 20 degrees.

Degrees? But we need to use radians!

Let us estimate as best we can:

20 × \frac{π}{180} = \frac{π}{9} ≈ 3.1416 × 0.11... ≈ 0.35 radians

Now, using just one extra term:

**sin x** = x − \frac{x^{3}}{3!} ...

**sin x** ≈ 0.35 − \frac{0.35^{3}}{3!} ≈ 0.35 − \frac{0.35*0.35*0.35}{6} **≈ 0.3428** (after much effort)

(Later when you get home you use a calculator to get **sin(20°) = 0.3420201...**, not bad!)

## Uses

These approximations are very useful in **astronomy **where many angles are very small.

Also in some areas of engineering and optics too.