Part 3/6:
The derivative of ( \sin(\theta) ) is ( \cos(\theta) ), and at ( \theta = 0 ), ( \cos(0) = 1 ).
The derivative of ( \tan(\theta) ) is ( \sec^2(\theta) ), which also equals 1 at ( \theta = 0 ).
Thus, in both cases, the linear approximation yields:
$$ L(\theta) = \theta $$
This simple relationship greatly facilitates calculations in fields like optics and lens design, where precision is vital.
Cosine Function
In contrast, for the cosine function, the linear approximation behaves differently:
- ( \cos(\theta) \approx 1 ) when ( \theta ) is near zero.
The derivative of ( \cos(\theta) ) is ( -\sin(\theta) ), which equates to zero at ( \theta = 0 ). Hence, for the cosine function, the linear approximation simplifies to:
$$ L(\theta) = f(0) = 1 $$