Question 790340: Explain why Sine (theta + pi/2) = Cosine(theta) and why Cosine (theta + pi/2) = -Sine(theta)
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Explain why Sine (theta + pi/2) = Cosine(theta) and why Cosine (theta + pi/2) = -Sine(theta)
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sin(x+π/2)=cos(x)
cos(x+π/2)=-sin(x)
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Adding π/2 moves the reference angle one quadrant over and changes the reference angle to the complement of the original angle. So, cos of the complement=sin of the angle. Also, the sign changes when moving the sin but the sign does not change when moving the cos. Four examples follow using degrees: (special angles for used for ease of explaning, but should work for any angles)
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let x=30 deg in quadrant I
reference angle=30 deg
30+90=120 deg in quadrant II
reference angle=60 deg
sin 30=1/2
cos 120=-1/2
-cos(x+90)=sin(x)
cos(x+90)=-sin(x)
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let x=60 deg in quadrant I
reference angle=60 deg
60+90=150 deg in quadrant II
reference angle=30 deg
cos 60=1/2
sin 150=1/2
sin(x+90)=cos(x)
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let x=120 deg in quadrant II
reference angle=30 deg
120+90=210 deg in quadrant III
reference angle=60 deg
cos 120=-1/2
sin 210=-1/2
sin(x+90)=cos(x)
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let x=330 deg in quadrant IV
reference angle=30 deg
330+90=420 deg in quadrant I
reference angle=60 deg
sin 330=-1/2
cos 420=1/2
cos(x+90)=-sin(x)
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