trigonometry.scad

LibFile: trigonometry.scad

Trigonometry shortcuts for people who can't be bothered to remember all the function relations, or silly acronyms like SOHCAHTOA.

To use, add the following lines to the beginning of your file:

include <BOSL2/std.scad>

File Contents

1. Section: 2D General Triangle Functions

   • law_of_cosines() – Applies the Law of Cosines for an arbitrary triangle.
   • law_of_sines() – Applies the Law of Sines for an arbitrary triangle.

2. Section: 2D Right Triangle Functions

   • hyp_opp_to_adj() – Returns the adjacent side length from the lengths of the hypotenuse and the opposite side.
   • hyp_ang_to_adj() – Returns the adjacent side length from the length of the hypotenuse and the angle.
   • opp_ang_to_adj() – Returns the adjacent side length from the length of the opposite side and the angle.
   • hyp_adj_to_opp() – Returns the opposite side length from the lengths of the hypotenuse and the adjacent side.
   • hyp_ang_to_opp() – Returns the opposite side length from the length of the hypotenuse and the angle.
   • adj_ang_to_opp() – Returns the opposite side length from the length of the adjacent side and the angle.
   • adj_opp_to_hyp() – Returns the hypotenuse length from the lengths of the adjacent and opposite sides.
   • adj_ang_to_hyp() – Returns the hypotenuse length from the length of the adjacent and the angle.
   • opp_ang_to_hyp() – Returns the hypotenuse length from the length of the opposite side and the angle.
   • hyp_adj_to_ang() – Returns the angle from the lengths of the hypotenuse and the adjacent side.
   • hyp_opp_to_ang() – Returns the angle from the lengths of the hypotenuse and the opposite side.
   • adj_opp_to_ang() – Returns the angle from the lengths of the adjacent and opposite sides.

Section: 2D General Triangle Functions

Function: law_of_cosines()

Synopsis: Applies the Law of Cosines for an arbitrary triangle.

Topics: Geometry, Trigonometry, Triangles

See Also: law_of_sines()

Usage:

• C = law_of_cosines(a, b, c);
• c = law_of_cosines(a, b, C=);

Description:

Applies the Law of Cosines for an arbitrary triangle. Given three side lengths, returns the angle in degrees for the corner opposite of the third side. Given two side lengths, and the angle between them, returns the length of the third side.

Figure 1.1.1:

Arguments:

By Position	What it does
a	The length of the first side.
b	The length of the second side.
c	The length of the third side.

By Name	What it does
C	The angle in degrees of the corner opposite of the third side.

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Function: law_of_sines()

Synopsis: Applies the Law of Sines for an arbitrary triangle.

Topics: Geometry, Trigonometry, Triangles

See Also: law_of_cosines()

Usage:

• B = law_of_sines(a, A, b);
• b = law_of_sines(a, A, B=);

Description:

Applies the Law of Sines for an arbitrary triangle. Given two triangle side lengths and the angle between them, returns the angle of the corner opposite of the second side. Given a side length, the opposing angle, and a second angle, returns the length of the side opposite of the second angle.

Figure 1.2.1:

Arguments:

By Position	What it does
a	The length of the first side.
A	The angle in degrees of the corner opposite of the first side.
b	The length of the second side.

By Name	What it does
B	The angle in degrees of the corner opposite of the second side.

---

Section: 2D Right Triangle Functions

This is a set of functions to make it easier to perform trig calculations on right triangles. In general, all these functions are named using these abbreviations:

• hyp: The length of the Hypotenuse.
• adj: The length of the side adjacent to the angle.
• opp: The length of the side opposite to the angle.
• ang: The angle size in degrees.

If you know two of those, and want to know the value of a third, you will need to call a function named like AAA_BBB_to_CCC(). For example, if you know the length of the hypotenuse, and the length of the side adjacent to the angle, and want to learn the length of the side opposite to the angle, you will call opp = hyp_adj_to_opp(hyp,adj);.

Figure 2.1:

Function: hyp_opp_to_adj()

Alias: opp_hyp_to_adj()

Synopsis: Returns the adjacent side length from the lengths of the hypotenuse and the opposite side.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• adj = hyp_opp_to_adj(hyp,opp);
• adj = opp_hyp_to_adj(opp,hyp);

Description:

Given the lengths of the hypotenuse and opposite side of a right triangle, returns the length of the adjacent side.

Arguments:

By Position	What it does
hyp	The length of the hypotenuse of the right triangle.
opp	The length of the side of the right triangle that is opposite from the primary angle.

Example 1:

include <BOSL2/std.scad>
hyp = hyp_opp_to_adj(5,3);  // Returns: 4

---

Function: hyp_ang_to_adj()

Alias: ang_hyp_to_adj()

Synopsis: Returns the adjacent side length from the length of the hypotenuse and the angle.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• adj = hyp_ang_to_adj(hyp,ang);
• adj = ang_hyp_to_adj(ang,hyp);

Description:

Given the length of the hypotenuse and the angle of the primary corner of a right triangle, returns the length of the adjacent side.

Arguments:

By Position	What it does
hyp	The length of the hypotenuse of the right triangle.
ang	The angle in degrees of the primary corner of the right triangle.

Example 1:

include <BOSL2/std.scad>
adj = hyp_ang_to_adj(8,60);  // Returns: 4

---

Function: opp_ang_to_adj()

Alias: ang_opp_to_adj()

Synopsis: Returns the adjacent side length from the length of the opposite side and the angle.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_hyp()

Usage:

• adj = opp_ang_to_adj(opp,ang);
• adj = ang_opp_to_adj(ang,opp);

Description:

Given the angle of the primary corner of a right triangle, and the length of the side opposite of it, returns the length of the adjacent side.

Arguments:

By Position	What it does
opp	The length of the side of the right triangle that is opposite from the primary angle.
ang	The angle in degrees of the primary corner of the right triangle.

Example 1:

include <BOSL2/std.scad>
adj = opp_ang_to_adj(8,30);  // Returns: 4

---

Function: hyp_adj_to_opp()

Alias: adj_hyp_to_opp()

Synopsis: Returns the opposite side length from the lengths of the hypotenuse and the adjacent side.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• opp = hyp_adj_to_opp(hyp,adj);
• opp = adj_hyp_to_opp(adj,hyp);

Description:

Given the length of the hypotenuse and the adjacent side, returns the length of the opposite side.

Arguments:

By Position	What it does
hyp	The length of the hypotenuse of the right triangle.
adj	The length of the side of the right triangle that is adjacent to the primary angle.

Example 1:

include <BOSL2/std.scad>
opp = hyp_adj_to_opp(5,4);  // Returns: 3

---

Function: hyp_ang_to_opp()

Alias: ang_hyp_to_opp()

Synopsis: Returns the opposite side length from the length of the hypotenuse and the angle.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• opp = hyp_ang_to_opp(hyp,ang);
• opp = ang_hyp_to_opp(ang,hyp);

Description:

Given the length of the hypotenuse of a right triangle, and the angle of the corner, returns the length of the opposite side.

Arguments:

By Position	What it does
hyp	The length of the hypotenuse of the right triangle.
ang	The angle in degrees of the primary corner of the right triangle.

Example 1:

include <BOSL2/std.scad>
opp = hyp_ang_to_opp(8,30);  // Returns: 4

---

Function: adj_ang_to_opp()

Alias: ang_adj_to_opp()

Synopsis: Returns the opposite side length from the length of the adjacent side and the angle.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• opp = adj_ang_to_opp(adj,ang);
• opp = ang_adj_to_opp(ang,adj);

Description:

Given the length of the adjacent side of a right triangle, and the angle of the corner, returns the length of the opposite side.

Arguments:

By Position	What it does
adj	The length of the side of the right triangle that is adjacent to the primary angle.
ang	The angle in degrees of the primary corner of the right triangle.

Example 1:

include <BOSL2/std.scad>
opp = adj_ang_to_opp(8,45);  // Returns: 8

---

Function: adj_opp_to_hyp()

Alias: opp_adj_to_hyp()

Synopsis: Returns the hypotenuse length from the lengths of the adjacent and opposite sides.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• hyp = adj_opp_to_hyp(adj,opp);
• hyp = opp_adj_to_hyp(opp,adj);

Description:

Given the length of the adjacent and opposite sides of a right triangle, returns the length of the hypotenuse.

Arguments:

By Position	What it does
adj	The length of the side of the right triangle that is adjacent to the primary angle.
opp	The length of the side of the right triangle that is opposite from the primary angle.

Example 1:

include <BOSL2/std.scad>
hyp = adj_opp_to_hyp(3,4);  // Returns: 5

---

Function: adj_ang_to_hyp()

Alias: ang_adj_to_hyp()

Synopsis: Returns the hypotenuse length from the length of the adjacent and the angle.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• hyp = adj_ang_to_hyp(adj,ang);
• hyp = ang_adj_to_hyp(ang,adj);

Description:

For a right triangle, given the length of the adjacent side, and the corner angle, returns the length of the hypotenuse.

Arguments:

By Position	What it does
adj	The length of the side of the right triangle that is adjacent to the primary angle.
ang	The angle in degrees of the primary corner of the right triangle.

Example 1:

include <BOSL2/std.scad>
hyp = adj_ang_to_hyp(4,60);  // Returns: 8

---

Function: opp_ang_to_hyp()

Alias: ang_opp_to_hyp()

Synopsis: Returns the hypotenuse length from the length of the opposite side and the angle.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj()

Usage:

• hyp = opp_ang_to_hyp(opp,ang);
• hyp = ang_opp_to_hyp(ang,opp);

Description:

For a right triangle, given the length of the opposite side, and the corner angle, returns the length of the hypotenuse.

Arguments:

By Position	What it does
opp	The length of the side of the right triangle that is opposite from the primary angle.
ang	The angle in degrees of the primary corner of the right triangle.

Example 1:

include <BOSL2/std.scad>
hyp = opp_ang_to_hyp(4,30);  // Returns: 8

---

Function: hyp_adj_to_ang()

Alias: adj_hyp_to_ang()

Synopsis: Returns the angle from the lengths of the hypotenuse and the adjacent side.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• ang = hyp_adj_to_ang(hyp,adj);
• ang = adj_hyp_to_ang(adj,hyp);

Description:

For a right triangle, given the lengths of the hypotenuse and the adjacent sides, returns the angle of the corner.

Arguments:

By Position	What it does
hyp	The length of the hypotenuse of the right triangle.
adj	The length of the side of the right triangle that is adjacent to the primary angle.

Example 1:

include <BOSL2/std.scad>
ang = hyp_adj_to_ang(8,4);  // Returns: 60 degrees

---

Function: hyp_opp_to_ang()

Alias: opp_hyp_to_ang()

Synopsis: Returns the angle from the lengths of the hypotenuse and the opposite side.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• ang = hyp_opp_to_ang(hyp,opp);
• ang = opp_hyp_to_ang(opp,hyp);

Description:

For a right triangle, given the lengths of the hypotenuse and the opposite sides, returns the angle of the corner.

Arguments:

By Position	What it does
hyp	The length of the hypotenuse of the right triangle.
opp	The length of the side of the right triangle that is opposite from the primary angle.

Example 1:

include <BOSL2/std.scad>
ang = hyp_opp_to_ang(8,4);  // Returns: 30 degrees

---

Function: adj_opp_to_ang()

Alias: opp_adj_to_ang()

Synopsis: Returns the angle from the lengths of the adjacent and opposite sides.

Topics: Geometry, Trigonometry, Triangles

See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp()

Usage:

• ang = adj_opp_to_ang(adj,opp);
• ang = opp_adj_to_ang(opp,adj);

Description:

For a right triangle, given the lengths of the adjacent and opposite sides, returns the angle of the corner.

Arguments:

By Position	What it does
adj	The length of the side of the right triangle that is adjacent to the primary angle.
opp	The length of the side of the right triangle that is opposite from the primary angle.

Example 1:

include <BOSL2/std.scad>
ang = adj_opp_to_ang(sqrt(3)/2,0.5);  // Returns: 30 degrees

---