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Math 150: 5.4 Geometric Proof for cos α − β Expansion Identity
Objective: Students will derive the basic sum and difference trig identity
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cos α − β = cos α cos β + sin α sin β starting from analytical geometry principles.
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respectively (where α > β ). What are the rectangular ( x , y ) coordinates of A and B?
1. Points A and B lie on the unit circle rotated from 1, 0 through angles of α and β
2. Plot point A in quadrant III close to the negative y –axis and plot point B in quadrant
III close to the negative x- axis on the unit circle above.
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= 2 − 2cos α cos β − 2sin α sin β .
3. Use the sketch to prove that d AB
© Raelene Dufresne 2013
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Math 150: 5.4 Geometric Proof for cos α − β Expansion Identity
4. On the second unit circle below, plot your points A and B. On the first, plot the
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points Q 1, 0 and P cos θ, sin θ where θ = α − β . (So, in what quadrant does θ lie?)
5. Sketch the line segments AB and PQ on the grids above. What is true about d AB
and d PQ ?
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6. Prove that cos α − β = cos α cos β + sin α sin β using your true statement in #5. (This
proof will not be a LS-RS proof, but it is another version of a valid proof.)
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