If , use trigonometric substitution to write as a trigonometric function of $\theta$, where .

Question

If  , use trigonometric substitution to write  as a trigonometric function of $\theta$, where  .

Quizplus · Accepted Answer

Let F1F1F1S1F1F1F10=φ.
Using the Pythagorean Theorem, we have:
F1F1F1S11=√(S1S1S1O2+F1F1F1O2)=√(1+cos2φ).
Substituting $√(1+cos^2φ)=secφ+tanφ$, we have:
11eb80b1_afd1_538f_a291_4354c5e8d7b3_TB8633_11=∫secφ+tanφdφ=ln|secφ+tanφ|+C.
Taking the antiderivative gives us:
11eb80b1_afd1_5390_a291_dd7dc1155f98_TB8633_11=ln|tan(φ/2)+sec(φ/2)|+C.
Using the identity $sec(φ/2)=√[(1+cosφ)/2]$ and $tan(φ/2)=sinφ/(1+cosφ)$, we have:
11eb80b1_afd1_5390_a291_dd7dc1155f98_TB8633_11=ln|sinφ+√(1+cosφ)|+C.
Finally, substituting φ=F1F1F1S1F1F1F10 and 11eb80b1_afd1_5391_a291_77770b311d43_TB8633_11=0, we have:
0=ln|sin(F1F1F1S1F1F1F10)+√(1+cos(F1F1F1S1F1F1F10))|+C.
Since 11eb80b1_afd1_5391_a291_77770b311d43_TB8633_11=0, we can solve for C and simplify the expression to get:
11eb80b1_afd1_5390_a291_dd7dc1155f98_TB8633_11=ln|tan(φ/2)|=ln|tan(F1F1F1S1F1F1F10/2)|.
Thus, the answer is B.

If , Use Trigonometric Substitution to Write as a Trigonometric