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 − θ22
So let us try the better version of 1 − θ22 :
| θ (radians) | cos θ | 1 − θ22 | (1−θ22) − 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 − θ22 ?
The secret is the Taylor Series expansion of cos:
cos x = 1 − x22! + x44! − ...
So ... we can use more terms if want more accuracy!
Likewise we can improve sine:
sin x = x − x33! + x55! − ...
Or tan, or other functions like ex
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 × π180 = π9 ≈ 3.1416 × 0.11... ≈ 0.35 radians
Now, using just one extra term:
sin x = x − x33! ...
sin x ≈ 0.35 − 0.3533! ≈ 0.35 − 0.35*0.35*0.356 ≈ 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.