Trigonometry on the job: angles, rise and run

The Pythagorean theorem connects the three sides of a right triangle, but it says nothing about the angles. Trigonometry fills that gap. With it, a tradesperson who knows one side and one angle can find every other side, and one who knows two sides can find the angle. That covers bending conduit, setting a ramp or a pipe at an angle, working out how high a boom reaches, and checking a ladder's setup angle.

Trigonometry has a reputation for being difficult, but the job-site version needs only three ratios and a calculator. Sam Torres uses it every time he bends conduit; Maya will use it for pipe offsets in lesson 8. This lesson builds the foundation for both.

Naming the sides

In a right triangle, pick one of the two angles that is not the right angle and call it the reference angle, usually written with the Greek letter theta. The sides are then named relative to that angle:

• The hypotenuse is the longest side, opposite the right angle, as in lesson 6.
• The opposite side is across from the reference angle; it does not touch it.
• The adjacent side touches the reference angle and is not the hypotenuse.

In trade language, if the reference angle sits at the bottom of a slope, the opposite side is the rise, the adjacent side is the run, and the hypotenuse is the travel or the length along the slope. Keeping those three words in mind makes trigonometry feel familiar.

Sine, cosine and tangent

The three trigonometric ratios compare pairs of sides. For any particular angle, each ratio has a fixed value no matter how big the triangle is. That is what makes them useful: a 30 degree slope has the same rise-to-run ratio whether it is a short pipe offset or a long ramp.

Many people remember these with the word SOH-CAH-TOA: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. In trade words: sine is rise over travel, cosine is run over travel, tangent is rise over run.

Note that sin 30° is exactly 0.5, so at 30 degrees the travel is exactly twice the rise. At 45 degrees, the rise and run are equal and tan 45° is exactly 1. These two facts are worth remembering because they are used constantly.

Making sure the calculator is in degrees

Calculators can measure angles in degrees or in radians, a unit used in higher mathematics. All trade work uses degrees. If sin 30 comes out as -0.988 instead of 0.5, the calculator is in radians. Look for a DEG indicator on the display, or a setting in a phone calculator's scientific mode. Check it at the start of every job; it is one of the most common causes of wildly wrong answers.

Finding a side

To find an unknown side, choose the ratio that links the side you know, the side you want and the angle. Write the ratio, then rearrange.

Finding an angle

When you know two sides and want the angle, use the inverse functions, written \(\sin^{-1}\), \(\cos^{-1}\) and \(\tan^{-1}\), or on some calculators asin, acos and atan. They answer the question: which angle has this ratio? On most calculators they are the second function of the sin, cos and tan keys.

Choosing the right ratio

Every trigonometry problem comes down to the same short procedure. Following it every time prevents most errors.

• Sketch the triangle and mark the right angle and the reference angle.
• Label the side you know and the side you want as opposite, adjacent or hypotenuse.
• Pick the ratio that contains both: SOH if opposite and hypotenuse, CAH if adjacent and hypotenuse, TOA if opposite and adjacent.
• Write the equation, rearrange, then calculate. Check that the hypotenuse is the longest side and that the angle makes sense.

Angles in conduit bending

Electricians bend conduit to standard angles because the multipliers are easy to remember and the bender has marks for them. An offset is made of two equal bends in opposite directions. The distance between the bends along the conduit is the offset depth times a multiplier, which is simply one over the sine of the bend angle. Lesson 8 develops this for both pipe and conduit, but the link to trigonometry is direct: the multiplier for 30 degrees is 1 / sin 30° = 2, and for 45 degrees it is 1 / sin 45° = 1.414.

Benders and pipe fittings work with whole-number angles chosen for convenience, so the trigonometry is usually already in a chart. Knowing where the chart comes from lets you handle an angle that is not on it, or check a chart that has been copied wrong.

Trigonometry in code

Python's math module works in radians, so a program has to convert degrees with math.radians before calling sin, cos or tan, and convert back with math.degrees after an inverse function. This is the same trap as a calculator left in radian mode, and the code makes the conversion visible.

import math

def rise_from(travel, angle_deg):
    return travel * math.sin(math.radians(angle_deg))

def run_from(travel, angle_deg):
    return travel * math.cos(math.radians(angle_deg))

def angle_from(rise, run):
    return math.degrees(math.atan2(rise, run))

print(f"ramp 20 ft at 12 deg: rise {rise_from(20, 12):.2f} ft")
print(f"boom 40 ft at 65 deg: reach out {run_from(40, 65):.1f} ft, up {rise_from(40, 65):.1f} ft")
print(f"ramp 30 in over 360 in: {angle_from(30, 360):.2f} deg")
print(f"ladder 12 ft up, 3 ft out: {angle_from(12, 3):.1f} deg")
print(f"wrong mode check: sin(30) in radians = {math.sin(30):.3f}")

ramp 20 ft at 12 deg: rise 4.16 ft
boom 40 ft at 65 deg: reach out 16.9 ft, up 36.3 ft
ramp 30 in over 360 in: 4.76 deg
ladder 12 ft up, 3 ft out: 76.0 deg
wrong mode check: sin(30) in radians = -0.988

The last line shows what happens if you forget the conversion: sin of 30 radians is about -0.988, a nonsense answer for a 30 degree slope. Spotting a result like that is exactly the kind of check this course encourages.

Angles, slopes and percent grades

The same slope can be described three ways: as an angle in degrees, as a ratio of rise to run, or as a percent grade. Tangent is the bridge between them, because tangent is rise over run.

Small slopes are easier to understand as grades or ratios; nobody describes a drain as falling at 1.19 degrees. Steep angles, like a ladder or a conduit bend, are easier in degrees. Lesson 9 uses grades and ratios for drains, ramps and stairs.

Digital angle finders and smart levels

Many tradespeople now carry a digital level or angle finder that reads in degrees, percent, or inches per foot. The tool does the trigonometry, but you still need to understand it, for two reasons. First, you must set the right unit; a reading of 2.0 means very different slopes in degrees and in percent. Second, the reading is only as good as the surface it sits on and its calibration. A quick check is to read a surface, turn the tool end for end on the same spot, and read again. The two readings should match; if they do not, the tool needs calibrating.

When you use any of these tools or formulas, write the unit next to every slope you record: 2 percent, 1 in 50, a quarter inch per foot or 1.15 degrees. A bare number on a drawing markup or a daily report is an invitation for someone to read it in the wrong unit.