Understanding Angles - Right Triangles and Trigonometry
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#### Right Triangles and Trigonometry

For a woodworker, being able to "solve" right triangles is an extremely important skill. Compound miters, multi-sided structures and a variety of other complex building projects can all be understood and calculatedÂ using right triangle trigonometry.

If the term trigonometryÂ causes you to suffer a sudden onset of "math anxiety," you'll be happy to know that theÂ trigonometry you need for woodworkingÂ isn't all that complicated.Â Â A basicÂ knowledge is all that's necessary to solveÂ just about anyÂ angle problem that will ever come up in woodworking.Â Below, we'llÂ reproduce a few well-know formulas and relationships fromÂ right triangleÂ trigonometryÂ that can really come in handy in woodworking.

#### The Pythagorean Theorem

The Pythagorean Theorem is a relation in geometry between the length of the three sides of a right triangle:

For any right triangle ABC,

AÂ² +Â BÂ² =Â CÂ²

where A andÂ B are the sides of the triangle that meet at a 90 degree angle.

Example: Using the Pythagorean Theorem to determineÂ the length (L) ofÂ a diagonal brace that projects 8'' out from the surface ofÂ a wall to support a shelf, and 12'' downÂ from the bottom surface of the shelf :

 LÂ² = 8Â² + 12Â² => LÂ² = 208 => sqrt(LÂ²) = sqrt(208) => L = 14.4222'' => L = (approx.) 14-27/64''

#### The Trigonometric Functions Sine, Cosine and Tangent

The Pythagorean Theorem comes in handy for calculatingÂ the dimensions of the sides of right triangles, but forÂ angle calculations,Â it's necessary to use trigonometric functions.Â Trigonometric functions describe the relationships between the sides of a right triangle:

Sin(x) = O/H
Cos(x) = A/H
Tan(x) = O/A

The relationships apply to all right triangles regardless of the size or proportions of the triangle.

Example: Above, we used the Pythagorean Theorem to calculate the length of a diagonalÂ shelf braceÂ based on the 8''Â projection of the brace out from the wall, and the 12'' projection of the brace down from the bottom of the shelf.

This time, instead of having the brace meet the wall at exactly 12'' below the shelf, we want theÂ diagonalÂ braceÂ toÂ tilt out from the wall at a 30 degree angle, but still project exactlyÂ 8'' out from the wall.

Because we know angleÂ wÂ (where the wall and the brace meet) and the lengthÂ ofÂ the side of the right triangle formed by the brace, the wall and the bottom of the shelf opposite angle w,Â we canÂ useÂ the sineÂ function to determine the length (L)Â of the brace:

 sin(w) = 8/L => sin(30) *Â L = 8 => .5 * L = 8 => L = 8/.5 => L = 16''

IfÂ we also needed to know the distance (D) down fromÂ the bottom surface of theÂ shelf to the bottom tip of the brace, we could use either the Pythagorean Theorem:

D = sqrt(16Â² - 8Â²) = 13.8564

or the tangent function:

D = 8/tan(30) = 13.8564 = approximately 13-57/64''

The angle s of the cut where the brace meets the shelf is simply the complement of angle w:

90 - 30 = 60 degrees

#### Calculating Trigonometric Function Values

To calculate the value of trigonometric function, you either need to useÂ a scientific calculator orÂ a table of trigonometric function values. A scientific calculator is a handy tool to have around the shop and worth the expense of adding to your tool collection.Â  You'll also find tables of trigonometric values on the Rockler website at Woodworking Math Formulas, Tables and Calculators.

#### Inverse Trigonometric Functions

In some cases, the value of the trigonometric function is the known quantity, and the angle that itÂ correspondsÂ to needs to be determined.Â  In those cases it is necessary to use the inverse trigonometric functions.Â  Inverse trigonometric functions are, simply stated, trigonometric functions in reverse.Â In other words, they calculate the angle that corresponds toÂ a given trigonometric function value.Â  The inverse ofÂ the trigonometric functions areÂ often referred to as the "arc" function - "arc sine," for example, refers to the inverse function of the sine function.

ForÂ the purposes of making woodworking, the inverse trigonometric functions can be defined:

x = arcsin[sin(x)]
x = arccos[cos(x)]
x = arctan[tan(x)]

For example,

sin(30) = .5
arcsin(.5) = 30
arcsin[sin(30)] = 30
sin[arcsin(.5)] = .5

Example: Above,Â we used the Pythagorean TheoremÂ to calculate the length of a diagonal braceÂ that projectsÂ 8''Â out fromÂ a wall to support a shelf, and 12'' down from the bottom of the shelf. The Pythagorean Theorem told us the length of the braceÂ (14.4222'')Â but nothing about the anglesÂ we need to cut to fit the brace against the bottom of the shelf, and up against the wall.

We'll use the inverseÂ trigonometric functions to find theÂ angleÂ s of the cutÂ at the shelfÂ end of the braceÂ and the angleÂ wÂ of theÂ cut where the brace will meet the wall.

 tan(w) = 8/12 => arctan[tan(w)] = arctan(8/12) => wÂ = arctan(.0667) => wÂ = 33.69 degrees

Alternatively, we could have used either the arc sine and sine functions with the 14.4222''Â length of the diagonal brace and the 8'' projection of the brace out from the wall, or similarly, the arc cosine and cosine function with theÂ length of the brace and the 12''Â projection of the braceÂ down fromÂ the shelf.

The angleÂ sÂ for the cut at theÂ end of the brace that meets the shelfÂ could also be calculated using inverse trigonometric functions, but it's easier to simply subtract angleÂ w from 90 degrees to find its complement:

90 degrees - 33.69 degreesÂ = 56.31 degrees = angle s

posted on August 15, 2013 by Rockler