Justify the small angle approximation for tangentAssume 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?

Question

Justify the small angle approximation for tangentAssume 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&#039;s Theorem but not the last one. How do we justify the last one?

Kendrick Jacobs · Accepted Answer

Explanation:For small x,       1          cos      ⁡              x              ≈      1          1      −                        x          2                2              ≈  1  +            x      2        2  .

glyperezrl · Accepted Answer

Step 1Fairly 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 2Where we used known approximations and the geometric series.

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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