Question

Find the exact value by using a half-angle identity.

sin(7pi/8)

Answer 1

http://www.purplemath.com/modules/idents.htm

Answer 2

` π=180 `

Answer 3

7×22.5 Or (1/2)×7×45

Answer 4

Go for it. Use your ½∠ sine formula

Answer 5

what's x?

Answer 6

In your case it would be 315 (and that x is over 2 )

Answer 7

I personally prefer ` sin(a+b) ` though -:( ... anyway ,

Answer 8

how did you get 315?

Answer 9

still need help?

Answer 10

@ByteMe yes, more help would be GREATLY appreciated!

Answer 11

@amistre64 @ParthKohli @jim_thompson5910

Answer 12

Use the identity


\[\Large \sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos(\theta)}{2}}\]

Answer 13

In your case,

\[\Large \theta = \frac{7\pi}{4}\]

because half of this angle is

\[\Large \frac{\theta}{2} = \frac{1}{2}*\frac{7\pi}{4}\]

\[\Large \frac{\theta}{2} = \frac{7\pi}{8}\]

Answer 14

so theta is 7pi/4?

Answer 15

yes

Answer 16

so I have sin((7pi/4)/2)=+-sqrt((1-cos(7pi/4)/2) left?

Answer 17

keep going

use the unit circle to evaluate cos(7pi/4)

Answer 18

and then simplify

Answer 19

sqrt(2)/2?

Answer 20

sin((7pi/4)/2)=+-sqrt((1-(sqrt(2)/2)/2)?

Answer 21

keep going and simplify

Answer 22

can I simplify sin(7pi/4)?

Answer 23

no don't worry about the left side right now

Answer 24

would I distribute the - to the sqrt(2)/2?

Answer 25

inside the square root, you can multiply every term by 2 to get

\[\Large \sin\left(\frac{\frac{7\pi}{4}}{2}\right) = \pm\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\]
\[\Large \sin\left(\frac{\frac{7\pi}{4}}{2}\right) = \pm\sqrt{\frac{2-\sqrt{2}}{4}}\]

What's next?

Answer 26

getting rid of the denominator (4)?

Answer 27

You can break up the square root

Answer 28

\[\Large \sqrt{\frac{A}{B}} = \frac{\sqrt{A}}{\sqrt{B}}\]

Answer 29

so it'd be +-(2-sqrt(2)) / sqrt4

Answer 30

close

Answer 31

+-sqrt4?

Answer 32

\[\Large \sin\left(\frac{\frac{7\pi}{4}}{2}\right) = \pm\sqrt{\frac{2-\sqrt{2}}{4}}\]

\[\Large \sin\left(\frac{\frac{7\pi}{4}}{2}\right) = \pm\frac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}\]

Answer 33

\[+-\frac{ \sqrt{2-\sqrt{2}} }{ 2 }\]

Answer 34

So that means

\[\Large \sin\left(\frac{7\pi}{8}\right) = \pm\frac{\sqrt{2-\sqrt{2}}}{2}\]

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Since sin(7pi/8) = 0.38268343236 we know that the final answer is positive, so,

\[\Large \sin\left(\frac{7\pi}{8}\right) = \frac{\sqrt{2-\sqrt{2}}}{2}\]