Trigonometric Substitution Trigonometric Identity

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities

In the expression a² − x², the substitution of a sin(θ) for x makes it possible to use the identity 1 − sin²θ = cos²θ.

In the expression a² + x², the substitution of a tan(θ) for x makes it possible to use the identity tan²θ + 1 = sec²θ.

Similarly, in x² − a², the substitution of sec(θ) for x makes it possible to use the identity sec² − 1 = tan².

1 Examples

In the integral

one may use

so that the integral becomes

(provided a > 0; if a < 0 then √a² would be |a|, which would differ from a).

For a definite integral, one must figure out how the bounds of integration change. For example, as x goes from 0 to a/2, then sin(θ) goes from 0 to 1/2, so θ goes from 0 to π/6. Then we have

In the integral

one may write

so that the integral becomes

(provided a > 0).

2 Substitutions that eliminate trigonometric functions

Substitution can be used to remove trigonometric functions. For instance,

(but be careful with the signs)

Example (see quintic of l'Hospital [1]):

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Topics: Trigonometric Substitution, Trigonometric Identity...

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Home > Trigonometric substitution

1 Examples

2 Substitutions that eliminate trigonometric functions

Topics: Trigonometric Substitution, Trigonometric Identity...