Arithmetic Formulas Examples And Worksheet With Answer Key - Dr. Neal, Wku Page 2
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6Dr. Neal, WKU
Addition/Subtraction Formulas
sin(A + B) = sin A cosB + cos A sin B
cos(A + B) = cos A cos B ! sin A sin B
sin(A ! B) = sin A cosB ! cos A sin B
cos(A ! B) = cos A cos B + sin A sin B
Double Angle Formulas
2
2
sin(2 A) = 2 sin A cos A
cos(2 A) = cos
A ! sin
A
Half-Angle Formulas
!
B
$
1' cosB
!
B
$
1+ cosB
sin
cos
#
& = +
#
& = ±
"
%
"
%
2
2
2
2
2
5
Example. Assume tan A =
, with angle A in Quadrant III, and csc B = !
, with angle
7
4
B in Quadrant IV.
(a) Use right-triangle trig to determine the sines and cosines of angles A and B .
(b) Find sin(A + B) , cos(A + B) , sin(A ! B) , cos(A ! B) .
(c) Determine what quadrants the angles A + B and A ! B lie in.
(d) Find sin(2 A) and cos(2 A) . Determine what quadrant the angle 2 A lies in.
(e) Find sin(B / 2) and cos(B / 2) . Determine what quadrant the angle B / 2 lies in.
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Solution. With A in III and tan A =
, both x and y are negative. So y = !2 , x = !7 ,
7
2
2
and z = 2
= 53 .
For B in IV, y is negative but x is positive.
Also
+ 7
1
4
.
sin B =
= !
csc B
5
2
2
5
! 4
= 3
–7
–2
!4
53
5
Angle A in Quadrant III
Angle B in Quadrant IV
2
7
4
3
and cos A = !
and cos B =
sin A = !
sin B = !
53
53
5
5
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