Trigonometric Substitution
When algebraic substitution fails on roots like , we turn to the geometry of triangles. Converting to angles often clears the root.
The Three Substitutions
When you see a square root of a sum or difference involving , use a trig substitution. The key is to draw the reference triangle so you can convert back to at the end.
Case 1: → Use
Why it works:
So .
- • (Opp/Hyp)
- • (Adj/Hyp)
- •
Case 2: → Use
Why it works:
So .
- • (Opp/Adj)
- • (Hyp/Adj)
- •
Case 3: → Use
Why it works:
So .
- • (Hyp/Adj)
- • (Opp/Adj)
- •
The Reference Triangle
Mapping x to Theta
Worked Example: Circle Area
Case 1: Sine Substitution
Evaluate .
We see with . Use Case 1: .
Use half-angle:
Draw the reference triangle: Since , we have:
- • Opposite =
- • Hypotenuse =
- • Adjacent = (Pythagorean)
Reading from triangle:
- •
- •
- •
Level Up Examples
Example A: Case 2 (Tangent)
Evaluate .
See with . Use Case 2: .
Since :
- • Opposite =
- • Adjacent =
- • Hypotenuse =
Read:
Example B: Case 3 (Secant)
Evaluate .
See with . Use Case 3: .
Since :
- • Hypotenuse =
- • Adjacent =
- • Opposite =
Read:
•
•
Example C: Completing the Square First
Evaluate .
Now it's form with , .
Since :
- • Opposite =
- • Hypotenuse =
So:
Example D: Definite Integral
Evaluate .
See . This is , so .
Use: →
Let :
Since :
- •
- •
From triangle:
•
•
Example E: x in Numerator
Evaluate .
Now with , .
Since :
- • Opposite =
- • Hypotenuse =
- • Adjacent =
Read:
•
•
Practice Quiz
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