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Appendix C C-87<br />
6. Show that the line with slope m passing through the point (1, 0) intersects<br />
the unit circle at the point P a 1 m2 2m<br />
1 m 2, 1 m b . 2<br />
7. In Figure E let A be the point (1, 0), P be the point a 1 m2<br />
and<br />
1 m , 2m<br />
2 1 m b , 2<br />
O be the origin, and let angle POA be u.<br />
1 m 2<br />
2m<br />
(a) Explain why cos u and sin u <br />
1 m 1 m .<br />
2 2<br />
(b) Prove: tan(u2) m.<br />
(c) Let Q be the point (1, 0) and explain why angle PQA equals u2.<br />
1 tan 2 (u2)<br />
2 tan(u2)<br />
(d) Prove: cos u and sin u <br />
1 tan 1 tan 2 (u2) .<br />
2 (u2)<br />
Comment: In the study of techniques of integration in calculus the results of<br />
this exercise are the basis for a substitution that transforms a rational expression<br />
in sines and cosines to a rational expression in a single variable m.<br />
Exercises 55 and 52 of Section 9.2 and Exercise 101 in the Review Exercises<br />
for Chapter 9* cover results related to those in Exercises 6 and 7 above.<br />
*Precalculus: A Problems-Oriented Approach, 7th edition
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