If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying that limx->∞ sin x / x = 1 and...; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term. |
If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying that limx->∞ sin x / x = 1 and then using the limit definition of the derivative and the addition theorems; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term. |
Notation: With trigonometric functions, we use the abbreviation sin2(x) for (sin(x))2.
These are easiest proven from the unit circle:
The quickest way to prove these is Euler's formula. The tangent formula follows from the other two.
These can be shown by substituting x = y in the addition theorems, and using the Pythagorean formula for the latter two. Or use de Moivres formula with n = 2.
Solve the third and fourth double angle formula for cos2(x) and sin2.(x).
Substitute x/2 for x in the power reduction formulas, then solve for cos(x/2) and sin(x/2).
These can be proven by expanding their right-hand-sides using the addition theorems.
Replace x by (x+y) / 2 and y by (x-y) / 2 in the Product-to-Sum formulas.
If the trigonometric functions are defined in terms of geometry, then their derivatives can be found by first verifying that limx->∞ sin x / x = 1 and then using the limit definition of the derivative and the addition theorems; if they are defined by their Taylor series, then the derivatives can be found by differentiating the power series term by term.
The rest of the trig functions can be differentiated using the above identities and the rules of differentiation, for instance
The integral identities can be found in Wikipedia's table of integrals.
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