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Showing content from https://dlmf.nist.gov/4.20 below:

§4.20 Derivatives and Differential Equations ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.20 Derivatives and Differential Equations 4.20.1 d d z ⁡ sin ⁡ z = cos ⁡ z , 4.20.2 d d z ⁡ cos ⁡ z = − sin ⁡ z , 4.20.3 d d z ⁡ tan ⁡ z = sec2 ⁡ z , 4.20.4 d d z ⁡ csc ⁡ z = − csc ⁡ z ⁢ cot ⁡ z , 4.20.5 d d z ⁡ sec ⁡ z = sec ⁡ z ⁢ tan ⁡ z , 4.20.6 d d z ⁡ cot ⁡ z = − csc2 ⁡ z , 4.20.7 dn d z n ⁡ sin ⁡ z = sin ⁡ ( z + 1 2 ⁢ n ⁢ π ) , 4.20.8 dn d z n ⁡ cos ⁡ z = cos ⁡ ( z + 1 2 ⁢ n ⁢ π ) .

With a ≠ 0 , the general solutions of the differential equations

4.20.9 d2 w d z 2 + a2 ⁢ w = 0 , 4.20.10 ( d w d z ) 2 + a2 ⁢ w2 = 1 , 4.20.11 d w d z − a2 ⁢ w2 = 1 ,

are respectively

4.20.12 w = A ⁢ cos ⁡ ( a ⁢ z ) + B ⁢ sin ⁡ ( a ⁢ z ) , 4.20.13 w = ( 1 / a ) ⁢ sin ⁡ ( a ⁢ z + c ) , 4.20.14 w = ( 1 / a ) ⁢ tan ⁡ ( a ⁢ z + c ) ,

where A , B , c are arbitrary constants.

For other differential equations see Kamke (1977, pp. 355–358 and 396–400).

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