10 - 3 double and half-angle formulas


There are many applications to science and engineering    related to light and sound.  Many of these require equations involving the sine and cosine of x, 2x, 3x and more.  Doubling the sin x will not give you the value of sin 2x.  Nor will taking half of sin x, give you sin (x/2).
We can develop the double angle formulas directly by using the addition formulas for sine, cosine and tangent.
Example:
sin 2x = Sin(x + x) = sin x cos x + sin x cos x = 2 sin x cos x
Similarly, you can find the cos 2x and tan 2x.
Here are the double angle formulas:

 
Double Angle Formulas
sin 2a = 2sina cosa
cos 2a = cos2a - sin2a
= 1 - 2sin2a
= 2cos2a - 1



 

Half-angle Formulas


Sample Problems
1)  Find the sin 2x if cos x = 3/5 and x is in quadrant I.
Solution:
sin 2x = 2 sin x cos x
Since cos x = 3/5, sin x = 4/5. (Basic trig)
Thus, sin 2x = 2(3/5)(4/5) = 24/25
2)  Find the exact value of tan 22.5o
Solution:
We can use the half angle formula, because we know the value of tan 45o.


3)  Simplify  2sin22x + cos 4x
Solution:
2sin22x + cos 2(2x)
= 2sin22x  + cos22x - sin22x
= sin22x + cos22x
= 1  (pythagorean identity)

4)  Verify that (sin x + cos x)2 = 1 + sin 2x
Solution:
Work with the left side:
Square the binomial
sin2x + 2sin x cos x + cos2x
= 1 + 2sin x cos x
1 + sin 2x
We used pythagorean relationship and double angle formula!!

5)  Find the exact value of   sin 22.5ocos 22.5o
Solution:
We need to match this with the double angle formula for sin
(1/2)(2sin 22.5 cos 22.5)
Now use the double angle formula:
(1/2)sin 45

                                                                   __
(1/2)(\/ 2 /2)
                                                                __
= \/2 / 4

6)  simplify    sin 2a/ (1 - cos 2a)
Solution:

7)  Verify that  csc 2x = (1/2)csc x sec x
Solution:
work with the left hand side.  Change to sin 2x


You should now be ready to solve some trig equations involving the formulas we have been talking about!!