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Dr. Neal, WKU

MATH 117

Arithmetic Formulas

Given angles A and B , there several important arithmetic formulas for the sine and

cosine of the angles A ± B , as well as angles such as ! A , 90º ! B , 2 A , and B / 2 . In each

case, we can obtain an exact formula for the result in terms of the sine and cosine of the

original angles A and B without knowing the actual values of the angles A and B .

Negative Angles

Because angles A and ! A create the same x -

coordinates on the unit circle, but negative y -

(x , y)

coordinates, we have

cos A = cos(! A)

A

! A

and

sin(! A) = ! sin A

(x ,! y)

Complimentary Angles

Because the x and y values are switched for

(y, x )

complimentary angles such as 30º and 60º, we have

sin(90 º ! A) = cos(A)

90º ! A

(x , y)

and

A

cos(90º ! A) = sin(A)

A

Supplementary Angles

Because supplementary angles, such as 30º

and 150º, have the same y -coordinates but

180º ! A

negative x -coordinates, we have

(! x, y)

(x , y)

sin(A) = sin(180º ! A)

A

A

and

cos(180º ! A) = ! cos( A)

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