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The Magic Calculator and sin(x+h)=sin(x)cos(h)+sin(h)cos(x) How simple geometry illuminates the black box. by Bryan Dorner Pacific Lutheran University

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2 Introducing the Sine and Cosine to Students

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3 Is Technology a Black Box? My old reply: Use your calculator. Hidden message: (It’s smart enough, even if you aren’t.)

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4 What I’ll Show Today How beginning Trig students can compute the sine and cosine of arbitrary angles. How the same ideas lead to an easy derivation of the sine addition formula. Case study in using technology to combine numeric, geometric and algebraic views in a meaningful computation.

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5 Computing the Sines and Cosines We show how to find the sine and cosine of any angle between two angles such as at P and Q whose sine and cosine are known. We first find sin(x) and cos(x) when x is exactly halfway between the angles.

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6 Averaging the coordinates of P and Q gives the coordinates of point A=(a1,a2) at the midpoint of the chord joining P and Q. We get an approximation to the coordinates of B, but we can do better.

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7 We can use the coordinates of A=(a1,a2) and similar triangles to compute the coordinates of B exactly! O A P = ( cos(x+h), sin(x+h) ) Q = ( cos(x-h), sin(x-h) ) B = ( cos(x), sin(x) ) h x Unit Circle

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8 Seeing the Whole Computation

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9 Finding Arbitrary Values Between Known Values Simply iterate the above procedure to "close in" on the desired angle. (This method is similar to the Bisection method for finding the root of a continuous function.) A TI-82 program or spreadsheet can be used to implement the procedure.

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10 Numerical Experiments Show: The formula works no matter which quadrant P and Q are in - unless h is too big. - indeed any time cos h is negative. If cos h is negative then the method produces values of the correct magnitude, but opposite sign. (Values are on opposite side of the circle.)

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11 Fixing the Formula When cos(h) is positive, When cos(h) is negative, So, works for any angles x and h (as long as cos(h) ≠ 0).

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12 Derivation of sin(x+h)=sin(x)cos(h)+cos(x)sin(h) (preliminary remark) so and this holds even when cos(h)=0. Recall that a2 was found by averaging the known sine values. So, sin(x) cos(h) =.5( sin(x+h) + sin(x-h) is valid for all values of x, h.

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13 Derivation of sin(x+h)=sin(x)cos(h)+cos(x)sin(h) (continued) Since, (1)sin(x) cos(h) =.5( sin(x+h) + sin(x-h) ) is valid for all values of x and h, it is valid when the values of x and h are interchanged: sin(h) cos(x) =.5( sin(h+x) + sin(h-x) ) (2) =.5( sin(x+h) - sin(x-h) ) ( using sin(-t)=-sin(t) ) Adding (1) and (2) gives: (3)sin(x) cos(h) + cos(x) sin(h) = sin(x+h) (As easy as 1,2, 3.)

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