Mathematical Tools for Physics

Basic Stuff

1.1 Trigonometry
The common trigonometric functions are familiar to you, but do you know some of the tricks to remember (or
to derive quickly) the common identities among them? Given the sine of an angle, what is its tangent? Given
its tangent, what is its cosine? All of these simple but occasionally useful relations can be derived in about two
seconds if you understand the idea behind one picture. Suppose for example that you know the tangent ofθ, what
issinθ? Draw a right triangle and designate the tangent ofθasx, so you can draw a triangle withtanθ=x/ 1.

x

1

q

The Pythagorean theorem says that the third side is

√

1 +x^2. You now read the
sine from the triangle asx/

√

1 +x^2 , so

sinθ=

tanθ

√

1 + tan^2 θ

Any other such relation is done the same way. You know the cosine, so what’s the cotangent? Draw a different
triangle where the cosine isx/ 1.

Radians
When you take the sine or cosine of an angle, what units do you use? Degrees? Radians? Other? And who
invented radians? Why is this the unit you see so often in calculus texts? That there are 360 ◦in a circle is
something that you can blame on the Sumerians, but where did this other unit come from?

R 2 R

s

θ

2 θ

1