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if the sine ratio is given, then how can sin 0/2 be determined?

Sagot :

Step-by-step explanation:

If you know, the sine. You must find the cos ratio, using the Pythagorean trig theorem.

Next, you use the half angle identity for sin

[tex] \sin( \frac{ \alpha }{2} ) = \sqrt{ \frac{1 - \cos(a) }{2} } [/tex]

Example.

[tex] \sin( \alpha ) = \frac{ \sqrt{3} }{2} [/tex]

We must find

[tex] \sin( \frac{ \alpha }{2} ) [/tex]

First, use the Pythagorean identity

[tex] \sin {}^{2} ( \alpha ) + \cos {}^{2} ( \alpha ) = 1[/tex]

[tex]( \frac{ \sqrt{3} }{2} ) {}^{2} + \cos {}^{2} (a) = 1[/tex]

[tex] \frac{3}{4} + \cos {}^{2} (a) = 1[/tex]

[tex] \cos {}^{2} (a) = \frac{1}{4} [/tex]

[tex] \cos( \alpha ) = \frac{1}{2} [/tex]

Now use the half angle identiy

[tex] \sin( \frac{ \alpha }{2} ) = \sqrt{ \frac{1 - \cos( \alpha ) }{2} } [/tex]

[tex] = \frac{ \sqrt{1 - \frac{1}{2} } }{ \sqrt{2} } [/tex]

[tex] = \frac{ \sqrt{ \frac{1}{2} } }{ \sqrt{2} } [/tex]

[tex] = \frac{ \sqrt{ \frac{1}{2} } }{ \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } = \frac{ \sqrt{1} }{ \sqrt{4} } = \frac{1}{2} [/tex]

So the answer for our example is 1/2

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