Trigonometry
Made Easy (Part 2)
Last
month we outlined a four-step process for making trig easy and
useful. We discussed
drawing the triangle on the part and labeling only the known and
needed sides. Now let’s review the three basic trig formulas and
selecting the correct rule.
Step
#1:
Draw a 90º triangle, this step is often the most difficult.
Step
#2:
Label two sides of the triangle, "the known
side and the needed
side".
Step
#3:
Select the correct trig rule
Step
#4:
Calculate the unknown.
There
are three basic trig rules, the sine rule, cosine rule and tangent
rule. The formulas for
each are shown here.
Sine
rule sine
of angle = opposite ÷ hypotenuse (SOH)
Cosine
rule cosine
of angle = adjacent ÷ hypotenuse
(CAH)
Tangent
rule tangent of angle = opposite
÷ adjacent (TOA)
Recall
that the opposite, adjacent and hypotenuse are lengths of specific
sides of the triangle. These
rules use lengths of sides and the size of one angle to determine
the unknown values. (Use
the helpful SOH-CAH-TOA to remember the formulas!)
As
with any mathematic formula, the rule can be shifted, depending on
the unknown value. If
you know the value of the opposite and hypotenuse, you can use the
sine rule to calculate the value of the angle.
Also, if you know the size of the angle and the length of the
hypotenuse, you can calculate the length of the adjacent side.
Use our helpful formula sheet to see the variations of each
rule, get it at www.cnc-training.com.
These
rules use the sides of the triangle and the value of the angle.
Each rule uses two sides of a triangle – so how do you know
which rule to use?
Selecting
the correct rule does not have to be difficult, the key is in
labeling the triangle. Remember,
we only label the known
and needed features of the triangle.
Let’s
look at our sample part. The
triangle was identified and the unknown angle is marked.
Let’s label the sides with what we know
and what we need.
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Using
the dimensions from the print, we can determine the length of the
adjacent side and the length of the opposite side.
Now the triangle is labeled with what we know and what we
need.
What
we know:
Length of the adjacent side (1.2510”)
Length of the opposite
side (0.8760”)
What
we need:
Size of marked angle
With
the triangle marked with the known and needed it is easy to select
the correct rule. Which
rule uses the adjacent and opposite
sides to calculate the length of the angle?
The
tangent rule uses these lengths to find the size of the angle.
It
is important to only label two sides, never
label all three sides. When three sides are labeled it is very
difficult to choose the correct rule.
When you label just two sides, by the process of elimination,
you have only one choice when selecting a trig rule.
In this case, the only rule that uses both the opposite and
the adjacent side is the tangent rule.
When
working with a turned part, the majority of trig work will use the
tangent rule. The angle
should be measured from the horizontal axis (centerline) so one side
represents the X axis (generally the opposite side and this is a
radius value) and the other side represents the Z axis (the adjacent
side).
Marking
the triangle correctly is the key to selecting the correct trig
rule. Review this
example and check back next month when we use program code to
calculate an angle value.
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