A lot of companies hire math majors. Jake can check with the placement office at the university he attends, and the profs, job fairs.

]]>That’s what I call it. Alternatively the expression “Fundamental Law of Trigonometry” might be applied to the statement –

SO(|R,2) = R/Z

]]>Trig has a fundamental law? Really?

]]>That is a nice proof. But aside from the details of any proof of the sine-cosine addition formulas these formulas have a meaning and significance in themselves. And this meaning and significance is exactly that the assignment of real numbers to rotations as angles makes the addition of real numbers correspond to the composition of rotations thus serving to reduce the study of the geometry of the plane to the study of the addition of real numbers.

The fundamental law of trigonometry is

Rotation(theta1 + theta2) = Rotation(theta2) o Rotation(theta1)

in words – “A rotation of angle theta1 followed by a rotation of angle theta2 is equal to a rotation of angle theta1 + theta2”.

The sine-cosine addition formulas are a somewhat disguised version of the above fundamental law. They are just the analytic equivalent of the fundamental law using the linearity of rotations.

]]>I’m going to have to look at both of these. From a teaching standpoint, I really like the unit circle, as it lets me beat some more algebra into their heads. Thanks for the additional ideas.

]]>Let us see things in a triangle (the starting point of trigonometry).

sinA, sinB are actually the lengths of the opposite sides of the angle A and B in the triangle ABC (just take 2R=1 in the law of sine).

Therefore, sinA cosB = length of side b times cos(adjacent angle) = projection of side b on side c.

sinAcosB+sinBcosA = sum of projections = length of side c = sinC

C= pi -A-B, we get sin(A+B)=sinAcosB+sinBcosA. QED.

I think this is the “real geometrical” proof of the addition formula

Remark: sinA originally is the opposite side of A, in a right triangle; in view of the law of sin, sinA IS the opposite side of A in any triangle, as long as we see things geometrically—-taking 2R=1 means an expansion of the figure and does NOT change the GEOMETRY of the whole picture.

Hope that you like the geometric proof (which I invented some years ago)

]]>