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Trigonometry (2013)

Page history last edited by Joe Redish 2 years ago

Class Content I >Modeling with mathematics > Math recap

2.2.6

 

Basic trig functions

We will be using trigonometry throughout this physics class. It plays particularly important roles in understanding motion, forces, and optics. Although trig can be quite complex, a few simple ideas are enough for most applications.  Here's the key:

 

If you change the size of a right triangle so that all the sides change by the same scale factor, then the ratio of any two sides stays the same.

 

That seems rather trivial.  Of course if you change two things by the same factor their ratio stays the same!  But that leads to an interesting way of saying it.

 

The ratio of any two sides in a right triangle only depends on the shape of the triangle.  Since the shape is a function of the angles in the triangle, the ratio of any two sides in a right triangle only depends on the angle.

 

Let's look at a specific -- if generic -- triangle: a right triangle in which one of the acute angles is θ.

We've labeled one of the non-right angles θ. We call the side opposite to the specified angle "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.

 

Our scaling conclusion 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(θ), cos(θ), and tan(θ) respectively.

sin(θ) = opposite/hypotenuse

cos(θ) = adjacent/hypotenuse

tan(θ) = opposite/adjacent.

 

That's most of what you need to know.

 

Implications

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

 

tan(θ) = sin(θ) / cos(θ).

 

And the Pythagorean Theorem, a2 + o2 = h2, implies

 

sin2(θ) + cos2(θ) = 1.

 

Radians

One more thing you need to know from trig is the definition of radian.  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, we should define a dimension, but since the mathematicians, who work without units (like trapeze artists working without a net) have come up with a "natural" definition of angle, we tend to use that.  This is a bit strange because we then have a unit without a dimension, but that is the convention.

 

The standard unit of angle that mathematicians have defined is the radian.  This is defined in the context of a circle.  If we consider an angle measured from the center of a circle of radius R, then it intercepts the circle to cut off a length that we will call L.  The angle, θ, is defined to have the magnitude


θ = L/R 

 

 

The measure is called radians. Since the circumference of the entire circle is 2πR,the total angle in an entire circle is 2πR/R = This means that 

360o = 2π radians.

 

It is useful to keep the unit on the 2π even though it is unitless (since it is the ratio of two distances) since there is an arbitrary choice involved.  Since you are likely to be more familiar with degrees than with radians, you need to keep in mind the conversions:

 

360o = 2π radians

180o =   π radians

90o = π/2 radians

60o = π/3 radians

45o = π/4 radians.

 

Follow-ons:

 

Joe Redish 8/21/11

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