Resolved: "Use half-angle identities to find an exact value..."

Anito49

Open question

2022-08-20

Use half-angle identities to find an exact value without a calculator.

{\cos 75}^{\circ}

Jonathan Bailey

Beginner2022-08-21Added 10 answers

Let

75^{\circ} = \frac{150^{\circ}}{2} :

{\cos 75}^{\circ} = \cos \frac{150^{\circ}}{2}

Use the half-angle identity for sine

\sin \frac{u}{2} = \pm \sqrt{\frac{1 + \cos u}{2}} w h e r e u = 150^{\circ}

. Since

75^{\circ}

is in Ql, then we take the positive identity:

= \sqrt{\frac{1 + {\cos 150}^{\circ}}{2}}

By special angles,

{\cos 150}^{\circ} = - \frac{\sqrt{3}}{2}

= \sqrt{\frac{1 + (- \frac{\sqrt{3}}{2})}{2}}

= \sqrt{\frac{\frac{2}{2} - \frac{\sqrt{3}}{2}}{2}}

= \sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{2}}

= \sqrt{\frac{2 - \sqrt{3}}{4}}

= \frac{\sqrt{2 - \sqrt{3}}}{2}

Result:

\frac{\sqrt{2 - \sqrt{3}}}{2}

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