Trigonometry

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1. Solving Right Triangles

1.1. Quick Summary (Right Triangles)

Trig Function Definitions

Function Definition
sin(α) opposite / hypotenuse
cos(α) adjacent / hypotenuse
tan(α) opposite / adjacent

Solving Right Triangles

Know Want Compute
α, adjacent opposite = adjacent * tan(α)
hypotenuse = adjacent / cos(α)
α, opposite adjacent = opposite / tan(α)
hypotenuse = opposite / sin(α)
α, hypotenuse adjacent = hypotenuse * cos(α)
opposite = hypotenuse * sin(α)

1.2. If Know One Other Angle, Know All Angles

If in a right triangle I know one angle in addition to the right angle, then I known all three angles (because they must all add up to 180). So if the angles are α, β, γ and γ = 90, then:

β = 180 - 90 - α

This is perhaps obvious, but often I need to be reminded of the obvious.

1.3. Right Triangle: Know an Acute angle and Its Adjacent Leg

For a concrete example, let's say I'm computing the dimensions of a taper for a machine part. Imagine that the base form is a cylinder, and from this a taper will rise. If the included angle of the taper is 3 degrees (which would give a non-releasing taper), and if I draw a sectional view of the taper, the acute angle for each right triangle defining the taper outside of the cylinder is 1.5 degree. Let's say also that I know how far along this cylinder the taper should extend; say 15mm

In other words, I know all of the angles (one is 90, one is 1, and the other is 180-90-1.5 = 88.5) and I know one side adjacent to the acute angle. As it happens, I won't need this third angle.

First (and typically most usefully) I want to know the length of the side opposite the angle. For the taper example, this would give the amount that the taper rises above the cylinder; the difference between the radius of the the taper at its large end and the radius of the cylinder there.

The trigonometric function which involves the two sides (legs) is the tangent function.

tan(α) = opposite / adjacent

Here I know the adjacent, and want to find the opposite, so I can rearrange this to:

adjacent * tan(α) = opposite

So in the taper example:

15 * tan(1.5) = opposite

15 * 0.0262 = 0.39

What is the one remaining side (the hypotenuse)? The cosine function involves the hypotenuse and a side I know:

cos(α) = adjacent / hypotenuse

Rearranging for the unknown:

hypotenuse * cos(α) = adjacent

hypotenuse = adjacent / cos(α)

Solving for the example:

hypotenuse = 15 / 0.997 = 15.0051 = 15.01

Intuitively, this checks out: A slight taper such as this won't rise much, and the hypotenuse of a very skinny triangle such as this isn't going to be much longer than the leg adjacent to its acute angle.

1.4. Right Triangle: Know an Acute angle and Its Opposite Leg

What if I don't know the adjacent leg, but do know the opposite? (In terms of the taper example, this would imply that I know how high/thick the taper should be, and I want to find out how long a taper I need to cut to achieve this.)

Again it is the tangent function which involves the opposite and adjacent sides.

tan(α) = opposite / adjacent

It's just a matter of solving for the adjacent (rather than solving for the opposite, as above):

adjacent * tan(α) = opposite

adjacent = opposite / tan(α)

So, in this taper example I'll assume that I want a taper to increase 0.25mm in radius. This means that the length of the taper will be:

0.25 / 0.0262 = 0.2501 = 9.54

which seems about right.

What about the hypotenuse? Just as, in the previous section, the cosine function related the side I knew (then, the adjacent side) and the hypotenuse, so here the sine function relates the side I know (now, the opposite) and the hypotenuse. So:

sin(α) = opposite / hypotenuse

As before, rearrange for the unknown:

hypotenuse * sine(α) = opposite

hypotenuse = opposite / sin(α)

Solving for the example:

hypotenuse = 0.25 / 0.0262 = 9.55

which is just a hair over the length of the taper (9.54), which makes sense.


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