Mathematics.

trigonometric identities

Trigonometric Substitution Identities

Trigonometry30 minDifficulty5 out of 10

You should know: trigonometric identities

Overview

Trigonometric substitution is a technique for simplifying algebraic expressions containing √(a²−x²), √(a²+x²), or √(x²−a²) by replacing x with a trigonometric function of a new variable θ. Each radical form is matched to the substitution that makes a Pythagorean identity collapse the expression under the square root into a single trig function, removing the radical entirely. The technique is the bridge between the Pythagorean identities (sin²θ+cos²θ=1, 1+tan²θ=sec²θ) and integral calculus, and it is the standard method for evaluating integrals whose integrand contains these three radical forms, as well as for simplifying geometric expressions such as chord lengths and areas that naturally involve a difference or sum of squares.

Intuition

Each substitution is chosen so that a Pythagorean identity turns the sum or difference under the radical into a perfect square. For √(a²−x²), setting x = a sinθ gives a² − a²sin²θ = a²(1−sin²θ) = a²cos²θ, whose square root is simply a cosθ (no radical left). For √(a²+x²), setting x = a tanθ gives a² + a²tan²θ = a²(1+tan²θ) = a²sec²θ, whose square root is a secθ. For √(x²−a²), setting x = a secθ gives a²sec²θ − a² = a²(sec²θ−1) = a²tan²θ, whose square root is a tanθ. In each case you can picture a right triangle with the substitution encoded in its sides: for x = a sinθ, the triangle has hypotenuse a, opposite side x, and adjacent side √(a²−x²) — reading the triangle back out lets you convert a final answer in θ back into an expression in x.

Formal Definition

Definition

For a constant a > 0, the radical form present in the expression determines the substitution:

a2−x2:x=asin⁡θ,a2−x2=acos⁡θ,−π2≤θ≤π2\sqrt{a^2 - x^2}: \quad x = a\sin\theta, \quad \sqrt{a^2-x^2} = a\cos\theta, \quad -\frac{\pi}{2}\le\theta\le\frac{\pi}{2}
Case 1: a² − x²
a2+x2:x=atan⁡θ,a2+x2=asec⁡θ,−π2<θ<π2\sqrt{a^2 + x^2}: \quad x = a\tan\theta, \quad \sqrt{a^2+x^2} = a\sec\theta, \quad -\frac{\pi}{2}<\theta<\frac{\pi}{2}
Case 2: a² + x²
x2−a2:x=asec⁡θ,x2−a2=atan⁡θ,0≤θ<π2\sqrt{x^2 - a^2}: \quad x = a\sec\theta, \quad \sqrt{x^2-a^2} = a\tan\theta, \quad 0\le\theta<\frac{\pi}{2}
Case 3: x² − a²

Worked Examples

  1. Solve for θ: x = 2sinθ = 1 gives sinθ = 1/2, so θ = π/6.

    sin⁡θ=12  ⟹  θ=π6\sin\theta = \frac{1}{2} \implies \theta = \frac{\pi}{6}

  2. Apply the identity √(a²−x²) = a cosθ with a=2: √(4−x²) = 2cos(π/6).

    4−1=2cos⁡ ⁣(π6)=2⋅32=3\sqrt{4-1} = 2\cos\!\left(\frac{\pi}{6}\right) = 2\cdot\frac{\sqrt3}{2} = \sqrt3

Answer: √(4−1) = √3 ≈ 1.7321, matching direct computation √3.

Practice Problems

Difficulty 3/10

For the radical √(25 − x²), which substitution should be used, and what does the radical simplify to in terms of θ?

Difficulty 4/10

Using x = 3secθ, simplify √(x² − 9) at x = 6, and check the answer directly.

Difficulty 6/10

A circular window has radius 3 m. Using the substitution x = 3sinθ, express the area of the quarter-disk (x from 0 to 3, under y = √(9−x²)) as an integral in θ and state its value, given that this area equals (1/4)πr².

Quiz

For an expression containing √(a² − x²), the correct substitution is:
With the substitution x = a secθ for √(x² − a²), the radical simplifies to:
The substitution x = a tanθ is used for radicals of the form:

Summary

  • Three radical forms match three substitutions: √(a²−x²) → x=a sinθ, √(a²+x²) → x=a tanθ, √(x²−a²) → x=a secθ.
  • Each substitution uses a Pythagorean identity to collapse the expression under the radical into a perfect square, removing the square root.
  • A reference right triangle with the substitution's sides lets you convert a final answer back from θ into x.

References

  1. WebsiteWikipedia — Trigonometric substitution