Chapter 3: Laws of Sines and Cosines

• $\mathbf{\text{Obtuse Angles}}$
• $\mathbf{\text{The Law of Sines}}$
• $\mathbf{\text{The Law of Cosines}}$

The first science developed by humans is probably astronomy. Before the invention of clocks and calendars, early people looked to the night sky to help them keep track of time. What is the best time to plant crops, and when will they ripen? On what day exactly do important religious festivals fall?

By tracking the motions of the stars, early astronomers could identify the summer and winter solstices and the equinoxes. The rising and setting of certain stars marked the hours of the night.

If we think of the stars as traveling on a dome above the Earth, we create the celestial sphere. Actually, of course, the Earth itself rotates among the stars, but for calculating the motions of heavenly objects, this model works very well.

Babylonian astronomers kept detailed records on the motion of the planets, and were able to predict solar and lunar eclipses. All of this required familiarity with angular distances measured on the celestial sphere.

To find angles and distances on this imaginary sphere, astronomers invented techniques that are now part of spherical trigonometry. The laws of sines and cosines were first stated in this context, in a slightly different form than the laws for plane trigonometry.