Chapter 8: More Functions and Identities

• $$\mathbf{\text{Sum and Difference Formulas}}$$
• $$\mathbf{\text{Inverse Trigonometric Functions}}$$
• $\mathbf{\text{The Reciprocal Functions}}$

Mapmakers have always faced an unavoidable challenge: It is impossible to translate the surface of a sphere onto a flat map without some form of distortion. Over the years, a variety of map projections have been developed to suit different uses.

The sixteenth century was an age of discovery, when explorers and merchants began sailing to distant and previously unknown lands. But at that time there was no reliable technology for navigation. Although more regions of the world were being mapped more accurately, a flat map by itself was not enough to help a sailor in the middle of the ocean.

In 1569, the Flemish cartographer Gerardus Mercator published a new map using what is known as a cylindrical projection. To imagine how a Mercator projection works, picture shining a light through a glass globe onto a piece of paper rolled into a cylinder and wrapped around the globe. The cylinder is tangent to the globe at its equator.