Question

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

Answer 1

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.