Question

Find the exact value of sin22.5 (using the half angle formula)

i have the answer from the book, but i don't understand how they got there

Answer 1

half angle formula is obtained by manipulating the cos(2a) formula

Answer 2

cos(2a) has 3 equal solutions:

cos^2 - sin^2

1 - 2sin^2

2cos^2 -1

Answer 3

cos(2a) = 1-2sin^2 is helpful in this case..

cos(2a) -1 = 2sin^2

cos(2a) -1
-------- = sin^2
2

Answer 4

yes, i just don't understand how they simplified this:

sqrt ((1 - [sqrt2]/2) / 2) it is supposed to become sqrt ((2 - sqrt2) / 4 )

Answer 5

they had steps, and it didn't explain how they got from step to step

Answer 6

ohhhh......

Answer 7

so you wanna see it simplified then right?

Answer 8

it just said "common denominator" and i couldn't figure out how to simplify it

Answer 9

i guess yes, i just want it explained really

Answer 10

sqrt(1 - cos(45) /2) = sin(22.5) right?

sqrt([1 - sqrt(2)/2]/2)

we can seperate the "square root" sign to cover the top and bottom of the fraction seperately... for example:

sqrt(a/b) = sqrt(a) / sqrt(b)

right?

Answer 11

let me write it out this way so I dont confuse meself:

sqrt(2)
1 - -----
2
-----------
2

Answer 12

2 sqrt(2)
-- - -------
2 2
-------------
2

Answer 13

2 - sqrt(2)
---------
2
------------
2

Answer 14

2-sqrt(2) 1
-------- x ---
2 2

Answer 15

2 - sqrt(2)
---------
4

Do you follow that?

Answer 16

hold on

Answer 17

i'm confused. i thought you multiplied

2-sqrt(2) 2
-------- x ---
2 1

because 2 is the reciprocal isnt it?

Answer 18

or am i completely wrong

Answer 19

lets take a look at a simpler example:

1/2
--- = 1/4 right?
2

Answer 20

whenever we divide by a fraction we turn it over an multiply instead.

1/2
--- = 1/2 * 1/2 = 1/4
2


1/3
--- = 1/3 * 1/2 = 1/6
2

Answer 21

7
--- = 7 * (3/1) = 21
1/3

Answer 22

ohhhhhh got it thnks!

Answer 23

I knew you would :)

Answer 24

thank you!