# Small Angle Approximations

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

sin θ ≈ θ
cos θ ≈ 1 − θ22
tan θ ≈ θ

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.