Question
Find the exact value of the expressi \( \tan \left(\arccos \left(\frac{\sqrt{3}}{2}\right)\right)=\square \) (Type an exact answer, using \( \pi \) and
Ask by Reid Medina. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The exact value is \( \frac{\sqrt{3}}{3} \).
Solution
Find the value of \( \tan(\arccos(\frac{\sqrt{3}}{2})) \).
Calculate the value by following steps:
- step0: Calculate:
\(\tan\left(\arccos\left(\frac{\sqrt{3}}{2}\right)\right)\)
- step1: Rewrite the expression:
\(\frac{\sqrt{1-\left(\frac{\sqrt{3}}{2}\right)^{2}}}{\frac{\sqrt{3}}{2}}\)
- step2: Subtract the numbers:
\(\frac{\sqrt{\frac{1}{4}}}{\frac{\sqrt{3}}{2}}\)
- step3: Simplify the root:
\(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\)
- step4: Multiply by the reciprocal:
\(\frac{1}{2}\times \frac{2}{\sqrt{3}}\)
- step5: Reduce the numbers:
\(1\times \frac{1}{\sqrt{3}}\)
- step6: Multiply:
\(\frac{1}{\sqrt{3}}\)
- step7: Multiply by the Conjugate:
\(\frac{\sqrt{3}}{\sqrt{3}\times \sqrt{3}}\)
- step8: Multiply the numbers:
\(\frac{\sqrt{3}}{3}\)
The exact value of \( \tan \left(\arccos \left(\frac{\sqrt{3}}{2}\right)\right) \) is \( \frac{\sqrt{3}}{3} \).
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Simplify this solution Bonus Knowledge
To find the exact value of \( \tan \left(\arccos \left(\frac{\sqrt{3}}{2}\right)\right) \), first determine the angle whose cosine is \( \frac{\sqrt{3}}{2} \). This angle is \( \frac{\pi}{6} \) or \( 30^\circ \). Now we calculate the tangent of this angle: \[ \tan\left(\frac{\pi}{6}\right) = \frac{\sin\left(\frac{\pi}{6}\right)}{\cos\left(\frac{\pi}{6}\right)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}. \] So, the exact value of \( \tan \left(\arccos \left(\frac{\sqrt{3}}{2}\right)\right) \) is \( \frac{1}{\sqrt{3}} \). Final answer: \[ \tan \left(\arccos \left(\frac{\sqrt{3}}{2}\right)\right) = \frac{1}{\sqrt{3}}. \]
