How do you find the exact value of sin ⁡ ( x 2 ) and cos ⁡ ( x 2 ) if cos ⁡ x = − 1 4 and π &lt; x &lt; 2 π?

Question

How do you find the exact value of   sin  ⁡  (      x    2    ) and   cos  ⁡  (      x    2    ) if   cos  ⁡  x  =  −      1    4   and   π  &amp;lt;  x  &amp;lt;  2  π?

autarhie6i · Accepted Answer

Use the trig identity:   cos  ⁡  x  =  1  −  2      sin          2        ⁡  (      x    2    )  −      1    4    =  1  −  2      sin          2        ⁡  (      x    2    )  →      sin          2        ⁡  (      x    2    )  =      5    8    =  0.625  sin  ⁡  x      /    2  =  −  0.79Next, use trig identity:       cos          2        ⁡  (      x    2    )  −      sin          2        ⁡  (      x    2    )  =  cos  ⁡  x      cos          2        ⁡  (      x    2    )  =  −      1    4    +      5    8    =      3    8    →  ∗            cos      ⁡      x        2    =  −  0.61  ∗Check      sin          2        ⁡  (      x    2    )  +      cos          2        ⁡  (      x    2    )  =      5    8    +      3    8    =      8    8    =  1

How do you find the exact value of sin (x/2) and cos (x/2) if cos x= - 1/4 and pi< x<2pi

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