Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

For a right angle, (rt), triangle with short sides of length a and b and a long side of length c,

**
c2
= a2
+ b2 **

Here we provide a demonstration of its truth by showing that all possible rt triangle shapes have sides that satisfy the theorem.

This is done by constructing all rt triangle shapes within and touching the Unit Circle, a circle with radius 1.

A short side

When the length of the side

As the length of

This range of angle provides for drawing all possible rt triangle shapes and all such triangles have their long side

The family of all possible rt triangles all have short sides, one of which coincides in length with the x coordinate of a point on the Unit Circle and the other of which coincides in length with that point's y coordinate.

For examples, hover your mouse over any of the sixteen base angles listed below. For each base angle chosen, a graph with the corresponding inscribed triangle will be drawn. The graphs show the base angle of the chosen triangle, the x, y coordinates of the triangle's vertex on the Unit Circle and the squares of those coordinates.

Angle | |||||||

0 | 6 | 12 | 18 | 24 | 30 | 36 | 42 |

48 | 54 | 60 | 66 | 72 | 78 | 84 | 90 |

*(Note.
In mathematics, all fundamental trigonometric identities are proved using the Pythagorean theorem.)*

Closely related to the Unit Circle and the theorem of Pythagoras are the two fundamental trigonometric functions, the sine of an angle, and the cosine of an angle.

For any of our family of inscribed triangles, the length of its short side that lies on the x-axis is taken as the value of the cosine of the base angle.

Similarly, the length of the short side that is parallel to the y-axis provides the sine of the base angle.

In terms of the points around the Unit Circle,
their x, y coordinates correspond to the cosine and sine of the angle between the
side ** c** and the x-axis, the base angle.

That angle is considered
herein to increase from the value 0.0o
when ** c** is aligned with the positive x-axis through positive
values as

(The direction of rotation and initial alignment that are assumed here are not fully standardized among the professions.)

The following graph shows two full cycles of the sine and cosine values versus the angle of rotation.

For more compact discussion call the Base Angle, (the angle between c and a), the angle B. Call the angle between **
c** and the other short side A where B = (90-A).

From the symmetry seen in the above graph, it should be clear that the sine of B is the cosine of A and vice-versa and thus we can write the identity that sin(A) = cos(90-A).

Other clear relationships are that sin(-A) = -sin(A) and that cos(-A) = cos(A). These relationships are valid not only for angles of the given triangles but also for all planar triangles.

If the hypotenuse of one of our family of triangles were scaled, up or down, from unity to some value r, its sides would remain in the original proportion but would then have lengths r times cos(A) and r times sin(A).

Thus given the hypotenuse
r and one of the angles G, (A or B), the length of the short side adjacent to that
angle can be found as: r*cos(G) and the other as: r*sin(G).
*(The * is used in Excel2000 to signify the operation
of multiplication.)*

Similarly the length of a short side and one of the angles A or B will allow the hypotenuse to be determined.

Returning again to our family of rt triangles, and noting the equality of side lengths to sine and cosine values, we conclude that: sin(A)2 + cos(A)2 = 1. If the triangle were scaled this relationship would not be affected as the r2's would factor out of both sides of the equation.

This identity is often used to find cos(X) given sin(X) and vice-versa.

The spreadsheet is employed to discover a close approximation to the ratio of a circle's circumference to its diameter.

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