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Fill in the blanks to complete each of the following theorem statements:

If ais a positive real number, and if x_____and u_____, then the substitution x=asinugives us a2-x2=_____.

Short Answer

Expert verified

If ais a positive real number, and if x-a,aand u-π2,π2 , then the substitution x=asinu gives us a2-x2=acosu.

Step by step solution

01

Step 1. Given information  

If ais a positive real number, and if role="math" localid="1653537491868" x_____and u_____, then the substitution x=asinu gives us a2-x2=_____.

02

Step 2. Filling in the blanks to complete the theorem statements 

If ais a positive real number, and if x-a,aand u-π2,π2, then the substitution x=asinugives us a2-x2=acosu.

By substituting for xand simplifying, we get the above answer valid under the conditions mentioned. That is, a2-x2=a2-asinu2

=a2-a2sin2u=a2(1-sin2u)=a2cos2u=acosu

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