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MATH 1113 PreCalculus

Day #30, Double and Half Angle Identities

If we consider the special cases where A = B, and then look at the formulas for cos(A+B), sin(A+B) and tan(A+B), we get the important (and easy to use) "double angle formulas:"

1.  The double-angle cosine formula(s)
 

i)  cos(2*A) = cos2(A) - sin2(A).

  This formula has the wonderful advantage of looking a lot like the Pythagorean Identity, with just a "-" in place of the "+".  If you add the squares you get 1, if you subtract the squares you get cos(2*A) (or -cos(2*A) if you subtract the other way.)

But because of the Pythagorean Identity, it is also possible (and useful) to write this identity in terms of a single trig variable (either sine or cosine):

ii)  cos(2*A) = (1 - sin2(A)) - sin2(A) = 1 - 2sin2(A)
iii)  cos(2*A) = cos2(A) - (1 - cos2(A)) = 2cos2(A) - 1.

2.  The double angle sine formula:
 

sin(2*A) = 2*sin(A)*cos(A).

3.  The double angle tangent formula:
 

tan(2*A) = (2*tan(A))/(1 - tan2(A)).

The two versions of the double angle cosine formula (ii) and (iii) have several important applications, especially when read the other way (that is, if you solve for sin2(A) or cos2(A).)

The book calls these the "power reducing formulas:"

1.  sin2(A) = (1 - cos(2*A))/2
2.  cos2(A) = (1+cos(2*A))/2

They find many uses in Calculus, and, by letting B = 2*A, so that A = B/2, they also give us the very important "half-angle" formulas (take the square root, then worry about the sign later):

Half-angle formulas:

1. sin(B/2) = sqrt[(1 - cos(B))/2]*
2. cos(B/2) = sqrt[(1 + cos(B))/2]*
3. tan(B/2) = sqrt[(1 - cos(B))/(1+cos(B))]*

*Always remember that you may need the "-" square root, depending on the relative quadrants of B and B/2.

Personally, I memorize one of the cosine double angle formulas (like cosine squared minus sine squared), and the sine double angle formula, and then remember that I can recreate any of the others with a little thought.  Again, there's no need to commit the whole list to memory.

Example:  Find the exact trig values of sine, cosine and tangent of pi/8 = 22.5o.

Answer:  Since 22.5 = 45/2, we'll use the half-angle formulas and work off of 45o.

1.  sin(22.5) = sqrt[(1 - sqrt(2)/2)/2] = sqrt[2 - sqrt(2)]/2
2.  cos(22.5) = sqrt[2+sqrt(2)]/2
3.  tan(22.5) = sqrt[(2 - sqrt(2))/(2+sqrt(2))]

By re-arranging the sum and difference formulas in other ways, yet more (and more complicated) formulas can be derived -- see the book for details.

 


For Practice, try p. 641 #9 - 21 odd, 41 - 49 odd, 77 - 83 odd (for these last, just look up the appropriate formula -- product to sum or sum to product)
 

 


Assignment: Do the CourseCompass problems from section 7.5
 

 


Assignment (2): Do the Show-Your-Work problems for Chapter 7.
 

 

 

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Page Last Modified: 25 March, 2007

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