The approximations

The approximations

As \theta becomes small:

• \sin\theta\approx\theta
• \cos\theta\approx1-\frac12\theta^2
• \tan\theta\approx\theta

These only work when working with radians - not degrees.

What is a small angle?

A ‘small’ angle is considered anything under half a radian.

Sine approximation

For a small angle \theta, when working with radians:

\sin\theta\approx\theta

Tangent approximation

For a small angle \theta, when working with radians:

\tan\theta\approx\theta

Cosine approximation

For a small angle \theta, when working with radians:

\cos\theta\approx1-\frac12\theta^2

Approximate the value of \cos0.4, then find the percentage error

• \cos(0.4)
• \approx1-\frac12(0.4)^2
• =1-\frac{0.16}2
• =1-0.08
• =0.92

The actual value is around 0.92106, so:

• Difference: 0.92106-0.92=0.00106
• \frac{0.00106}{0.92106}
• =0.00115
• =0.115\%