Assume that angle x is small, using small angle approximation, sin(x)=x; cos(x)=1-(x^2)/2; and tan(x)=x. I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?

Lilliana Livingston 2022-07-15 Answered
Justify the small angle approximation for tangent
Assume that angle x is small, using small angle approximation,
sin ( x ) = x
cos ( x ) = 1 x 2 2 ;
and tan ( x ) = x.
I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?
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Answers (2)

Kendrick Jacobs
Answered 2022-07-16 Author has 16 answers
Explanation:
For small x, 1 cos x 1 1 x 2 2 1 + x 2 2 .

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glyperezrl
Answered 2022-07-17 Author has 5 answers
Step 1
Fairly simple actually: tan ( x ) = sin ( x ) cos ( x ) x 1 x 2 2 = x ( 1 + x 2 2 + x 4 4 + ) x ( 1 + O ( x 2 ) )
Step 2
Where we used known approximations and the geometric series.

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