I calculated sin⁡75∘ as 122+322, but the answer is 2+64. What went wrong? I calculated the exact value of sin⁡75∘ as follows: sin⁡75∘=sin⁡(30∘+45∘) =sin⁡30∘cos⁡45∘+cos⁡30∘sin⁡45∘ =12⋅12+32⋅12 =122+322 The actual answer is 2+64

They’re the same value. Multiply the numerator and denominator of your answer by 2 to see why. 1+322=22⋅1+322=2+64 You can also use sin⁡45=cos⁡45=22 (rationalizing 12) to get the answer more easily.

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High School
Geometry
Trigonometry
Trigonometric equations and identities

poveli1e

Answered question

2022-01-27

I calculated ${\sin 75}^{\circ}$ as $\frac{1}{2 \sqrt{2}} + \frac{\sqrt{3}}{2 \sqrt{2}}$ , but the answer is $\frac{\sqrt{2} + \sqrt{6}}{4}$ . What went wrong?
I calculated the exact value of ${\sin 75}^{\circ}$ as follows:
${\sin 75}^{\circ} = \sin (30^{\circ} + 45^{\circ})$
$= \sin 30^{\circ} \cos 45^{\circ} + \cos 30^{\circ} \sin 45^{\circ}$
$= \frac{1}{2} \cdot \frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}}$
$= \frac{1}{2 \sqrt{2}} + \frac{\sqrt{3}}{2 \sqrt{2}}$
The actual answer is
$\frac{\sqrt{2} + \sqrt{6}}{4}$

Answer & Explanation

kumewekwah0

Beginner2022-01-28Added 14 answers

They’re the same value. Multiply the numerator and denominator of your answer by

\sqrt{2}

to see why.

\frac{1 + \sqrt{3}}{2 \sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}} \cdot \frac{1 + \sqrt{3}}{2 \sqrt{2}} = \frac{\sqrt{2} + \sqrt{6}}{4}

You can also use

\sin 45 = \cos 45 = \frac{\sqrt{2}}{2}

(rationalizing

\frac{1}{\sqrt{2}}

) to get the answer more easily.

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