Math 2414 Section 7 4 Trigonometric Substitutions If you are told a b and c are the three sides of a right triangle and you are given that a c 2 b 2 where should you place each side If you are told a b and c are the three sides of a right triangle and you are given that b c 2 a 2 where should you place each side If you are told a b and c are the three sides of a right triangle and you are given that c a 2 b 2 where should you place each side For now we can loosely state a rule for labeling For consistency when we know the two legs of a right triangle whose vertices are positioned as drawn above we will always choose the horizontal leg to be a numerical value and the vertical leg to include a variable If instead we know the hypotenuse and a leg we will label the vertical leg with a variable I will expect a triangle to be drawn and completely labelled for each of these types of problems While there are formulas that aid with these types of problems I find they do not aid in student understanding of the concept Therefore you will not be allowed to use them on exams or quizzes You may use them on the online homework Often we will rely on inverses to back substitute in these problems Also a relationship can be made between derivatives of some of the inverse trigonometric functions and the trigonometric substitution we attempt to make For these reasons it is beneficial to revisit some of our inverse trigonometric derivatives Some Derivatives of Inverse Trigonometric Functions d 1 sin 1 x dx 1 x2 d 1 sec 1 x dx x x2 1 d 1 tan 1 x dx 1 x 2 dx Evaluate 16 x 2 32 For trigonometric substitutions you will often find changing bounds on the definite integrals too difficult Consider doing the problem as an indefinite integral and then back substituting the correct variable 1 Evaluate dx 1 x 0 2 2 Evaluate x 2 dx 9 x2 4 x 2 4 x 5 dx x 2 Evaluate 1