What Is the Sine of 1°?
Lesson 18Topics
x The sine and cosine addition formulas.
x The “error term” in the Pythagorean theorem: the law of cosines.
Formulas
x addition formulas:
ż sin xy sin xcos ycos xsin y.
ż cos xy cos xcos ysinxsin y.
Changing yWRíyWKHVHUHDGͼXVLQJsin yy yysin and cos cos ͽDVIROORZV
ż sin xy sin xcos ycos xsin y.
ż cos xy cos xcos ysinxsin y.
Putting y = xLQWRWKH¿UVWHTXDWLRQVJLYHVWKHIROORZLQJ
ż sin 2 xxx 2 sin cos.
ż cos 2 xx x cos^22 sin.
ͼ7KLV¿QDOHTXDWLRQLVRIWHQDEEUHYLDWHGDVcos 2 xxx cos^22 sin .ͽ
x law of cosines: If a triangle has sides a, b, and c with angle x between sides a and b, then
cab ab x^222 2 cos.
Summary
It is not at all obvious how a calculator works out the sine and cosine values for angles different from the
familiar 0°, 30°, 45°, 60°, and 90°. In this lesson, we look at the sine and cosine addition formulas that allow us
to compute trigonometric values for a wider range of values. However, as we see, the sine of 1° remains out of
RXUUHDFK²DWOHDVWIRUWKLVOHYHORIZRUNLQJHRPHWU\:HQHHGFDOFXOXV