What is the length of the
hypotenuse?
c^2 = a^2 + b^2 = 5^2 + 12^2 = 25 + 144
c^2 = 169
1.12 - Trigonometric functions
You will often encounter trigonometric functions in physics. You need to understand the
basics of the sine, cosine and tangent, and their inverses: the arcsine, arccosine and
arctangent.
The illustration above depicts an angle and three sides of a triangle. The sine (sin) of
the angle ș (the Greek letter theta, pronounced “thay-tuh”) equals the ratio of the side
opposite the angle divided by the triangle's hypotenuse. “Opposite” means the leg
across from the angle, as the diagram reflects.
The cosine (cos) of șequals the ratio of the side of the triangle adjacent to the angle,
divided by the hypotenuse. “Adjacent” means the leg that forms one side of the angle.
Finally, the tangent (tan) of șequals the ratio of the opposite side divided by the
adjacent side.
These three ratios are constant for a given angle in a right triangle, no matter what the
size of the triangle. They are useful because you are often given information such as
the length of the hypotenuse and the size of an angle, and then asked to calculate one
of the legs of the triangle. For instance, if asked to calculate the opposite leg, you would
multiply the sine of the angle by the hypotenuse.
You may also be asked to use the arcsine, the arccosine or the arctangent. These are
often written as siní^1 , cosí^1 and taní^1. These are not the reciprocals of the sine, cosine
and tangent! Rather, they supply the size of the angle when the value of the
trigonometric function is known. For example, since sin 30° equals 0.5, the arcsine
of 0.5 (or siní^1 0.5) equals 30°. This is also often written as arcsin(0.5); arccos and
arctan are the abbreviations for arccosine and arctangent.
In the old days, scientists consulted tables for these trigonometric values. Today,
calculators and spreadsheets can calculate them for you.