Trigonometry

Basic trig functions

We use trigonometry throughout this physics class, especially to understand motion, forces, and optics. Although trig can be complex, a few simple ideas are enough for most of our applications. Here's the key:

If you change the size of a triangle so that all the sides change by the same scale factor ("similar triangles"), then the ratio of any two corresponding sides stays the same.

Basically the only example we will use trig for is the right triangle. Let's look at one in the figure at the right. Since the angles have to add up to 180^o, and one of the angles is fixed to be 90^o, the shape of a right triangle is completely determined by one of the acute angles. Let's pick one and call it $\theta$.

We call the side opposite to the angle $\theta$ "opposite" (label, $o$), the non-hypotenuse side of the triangle that makes part of the angle "adjacent" (label, $a$), and the hypotenuse we label $h$. Each of these symbols stands for the length of that side.

Applying our general principle at the top of the page to this triangle, we get

The ratio of any two sides in a triangle only depends on the shape of the triangle.  Since the shape of a right triangle only depends on $\theta$, the ratio of any side to any other is only a function of $\theta$.

This implies that the ratios $a/h$, $o/h$, and $a/o$ only depend on the angle $\theta$.  The names given to these three functions are $\sin{\theta}, \cos{\theta}$, and $\tan{\theta}$ respectively.

$\sin{\theta}$ = opposite/hypotenuse

$\cos{\theta}$ = adjacent/hypotenuse

$\tan{\theta}$ = opposite/adjacent.

That's most of what you need to know. (Many of you learned the mnemonic "soh-cah-toa" for this in high school.)

Other important relationships

Of course other things you know from geometry and algebra lead to interesting and useful relations.  Simple algebra implies

$$\tan{\theta}= \frac{\sin{\theta}}{{\cos{\theta}}}$$ 

And the Pythagorean Theorem, $a^2 + o^2 = h^2$, implies

$$\sin^2{\theta} + \cos^2{\theta} = 1$$

Radians

Since we can divide a circle up into angles in any way we want, there is an arbitrariness to the measure of angle.  The "360 degrees in a circle" is a historical number left over from the way the Babylonian liked to do calculations in the same way that a yard is a measure of the arm length of an old English king. 

Since we have an arbitrary choice of how many parts we will divide a circle into, we should define a dimensionality to help us keep track of which arbitrary choice we are using. But since the mathematicians, who work without units (like expert trapeze artists working without a net) have come up with a "natural" definition of angle, the radian, we tend to use that. Since it's the ratio of two lengths, physicists have accepted the mathematicians' definition as dimensionless. This is a bit strange, because we then have a unit without a dimension, but that is the convention. If we only used radians we'd have no problems, but we often like degrees. You'll have to be careful! The best idea is to keep the unit with your angle number (degrees or radians) in your calculation and check your units. (And be sure your calculator is set in the correct mode — degrees or radians — whenever you ask it to calculate a trig function!)

The radian is defined in the context of a circle. Consider an angle measured from the center of a circle of radius $R$. Its arms intercept the circle to cut off a length that we will call $L$.  The angle, $\theta$, is defined to have the magnitude

$$\theta = \frac{L}{R}$$

This measure is called radians. Since the circumference of the entire circle is 2πR,the total angle in an entire circle is $\frac{2\pi R}{R}= 2\pi$.  This means that a full circle is

360^o = 2$\pi$ radians.

It is useful to keep the unit on the 2$\pi$. Since you are likely to be more familiar with degrees than with radians, you need to keep in mind the conversions:

360^o = 2$\pi$ radians
180^o =   $\pi$ radians
90^o = $\pi$/2 radians
60^o = $\pi$/3 radians
45^o = $\pi$/4 radians.

Radians will become particularly valuable when we consider the properties of sines and cosines as functions (for example, in the small angle approximation) or when we generalize the use of sines and cosines to describe not triangles but oscillations. (See trig functions for any value of the argument.)

Joe Redish 8/21/11