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Can anybody show me how to do this half angle identity problem STEP BY STEP? I always seem to be missing something

Can anybody show me how to do this half angle identity problem STEP BY STEP? I always-example-1
User Leonti
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1 Answer

2 votes

Answer:
\bold{-(7\sqrt2)/(10)}

Explanation:

It is given that θ is between 270° and 360°, which means that θ is located in Quadrant IV ⇒ (x > 0, y < 0). Furthermore, the half-angle will be between 135° and 180°, which means the half-angle is in Quadrant II ⇒
cos\ (\theta)/(2) <0

It is given that sin θ =
-(7)/(25) ⇒ y = -7 & hyp = 25

Use Pythagorean Theorem to find "x":

x² + y² = hyp²

x² + (-7)² = 25²

x² + 49 = 625

x² = 576

x = 24

Use the "x" and "hyp" values to find cos θ:


cos\ \theta=(x)/(hyp)=(24)/(25)


Lastly, input cos θ into the half angle formula:


cos\bigg((\theta)/(2)\bigg)=\pm \sqrt{(1+cos\ \theta)/(2)}\\\\\\.\qquad \quad =\pm \sqrt{(1+(24)/(25))/(2)}\\\\\\.\qquad \quad =\pm \sqrt{((25)/(25)+(24)/(25))/(2)}\\\\\\.\qquad \quad =\pm \sqrt{((49)/(25))/(2)}\\\\\\.\qquad \quad =\pm \sqrt{(49)/(50)}\\\\\\.\qquad \quad =\pm (7)/(5\sqrt2)}\\\\\\.\qquad \quad =\pm (7)/(5\sqrt2)}\bigg((\sqrt2)/(\sqrt2)\bigg)\\\\\\.\qquad \quad =\pm (7\sqrt2)/(10)

Reminder: We previously determined that the half-angle will be negative.

User Hatted
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