Math Problem Statement

Find the exact value of cos(5π/12).

Solution

The problem asks for the exact value of cos(5π12)\cos \left( \frac{5\pi}{12} \right).

To solve this, let's break down 5π12\frac{5\pi}{12} into angles whose cosine values we know. We can use the sum or difference identities for cosine to find the exact value.

Step-by-Step Solution

  1. Rewrite 5π12\frac{5\pi}{12} as a sum: 5π12=3π12+2π12=π4+π6\frac{5\pi}{12} = \frac{3\pi}{12} + \frac{2\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}

  2. Use the Cosine Sum Identity: cos(π4+π6)=cosπ4cosπ6sinπ4sinπ6\cos \left( \frac{\pi}{4} + \frac{\pi}{6} \right) = \cos \frac{\pi}{4} \cos \frac{\pi}{6} - \sin \frac{\pi}{4} \sin \frac{\pi}{6}

  3. Substitute Values:

    • cosπ4=22\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}
    • cosπ6=32\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}
    • sinπ4=22\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}
    • sinπ6=12\sin \frac{\pi}{6} = \frac{1}{2}
  4. Calculate: cos(π4+π6)=22322212\cos \left( \frac{\pi}{4} + \frac{\pi}{6} \right) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} =6424= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} =624= \frac{\sqrt{6} - \sqrt{2}}{4}

Final Answer

The exact value is: cos(5π12)=624\cos \left( \frac{5\pi}{12} \right) = \frac{\sqrt{6} - \sqrt{2}}{4}

Would you like further details on how to apply trigonometric identities, or do you have other questions?

Additional Practice Questions:

  1. What is the exact value of cos(7π12)\cos \left( \frac{7\pi}{12} \right)?
  2. How can you use the sine sum identity to find sin(5π12)\sin \left( \frac{5\pi}{12} \right)?
  3. What is tan(5π12)\tan \left( \frac{5\pi}{12} \right) using sum identities?
  4. How do you simplify cos(3π4π6)\cos \left( \frac{3\pi}{4} - \frac{\pi}{6} \right)?
  5. Find the exact value of cos(π3+π4)\cos \left( \frac{\pi}{3} + \frac{\pi}{4} \right).

Tip: Always look for ways to rewrite angles in terms of known values, as this often simplifies complex trigonometric calculations.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities
Cosine Sum Identity

Formulas

Cosine sum identity: cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

Theorems

Cosine Sum Identity

Suitable Grade Level

Grade 11-12