Trig-substitutions → Trig Rule

The goal is to simplify expression by CONVERTING terms of x into simplified expression in terms of θ.

How to do this? We can identify some trigonometric patterns in the expression and apply the Pythagorean Theorem.

▶ Cheatsheet on previous note: Basic Integral Rules ▶ Cheatsheet on previous note: Basic Differential Rules ▶ Cheatsheet on previous note: All trig identities ▶ Back to previous note on: U-substitution → Chain Rule

▶ Practice at Khan academy: Trigonometric substitution

Refer to Khan academy: Introduction to trigonometric substitution

Example

Solve:

• In this expression, we can easily identify there is a "pythagorean-like terms":

  
• We can see x and 2 as two sides of a triangle:

  
• Once we identify the pattern, we can easily get some information out of it in terms of θ:

  
• From the information tan(θ)=x/2 we can get the relationship between x & θ, which is: x=2tan(θ)

• And now we can convert the original expression to be in terms of θ:

  

Example

Solve: ...