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Trigonometry made easy, Part 2: all the angles you'll need to successfully apply trig.

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 90[degrees] 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 of angle = opposite / hypotenuse (SOH)


cosine of angle = adjacent / hypotenuse (CAH)


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, available at


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.

Using the dimensions from the print, we can deter mine 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 label only 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.

Steve Rose is a professional trainer and president of RTSI, Solon, OH. Rosaleen Rose offers Internet website development. They can be reached by phone, 440.542.3066; e-mail; or on the web at
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Title Annotation:shop talk
Author:Rose, Steve
Publication:Tooling & Production
Date:Nov 1, 2005
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