Math Problem Statement

how is 4cos(pi/6) equal 2sqrt3 shouldnt it be 4/2 since cos(pi/6) = 1/2 then multiple by 4 ?

Solution

Let's clarify this step-by-step to make sure we are on the same page:

  1. You are working with the expression 4cos(π6)4 \cos\left( \frac{\pi}{6} \right).

  2. Step 1: Find cos(π6)\cos\left( \frac{\pi}{6} \right).

    • From trigonometric values, we know that cos(π6)=32\cos\left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}.
  3. Step 2: Multiply the result by 4: 4×cos(π6)=4×324 \times \cos\left( \frac{\pi}{6} \right) = 4 \times \frac{\sqrt{3}}{2}

  4. Step 3: Simplify the expression: 4×32=432=234 \times \frac{\sqrt{3}}{2} = \frac{4\sqrt{3}}{2} = 2\sqrt{3}

So the expression simplifies to 232\sqrt{3}.


Your confusion:

  • You mentioned that you thought cos(π6)=12\cos\left( \frac{\pi}{6} \right) = \frac{1}{2}, but actually cos(π6)=32\cos\left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}.
  • When you multiply by 4, you get 4×324 \times \frac{\sqrt{3}}{2}, not 4×124 \times \frac{1}{2}, hence the result is 232\sqrt{3}.

Would you like more details or have any questions on this? Here are some follow-up questions that might help deepen your understanding:

  1. Why does cos(π6)=32\cos\left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}?
  2. How can we visualize cos(π6)\cos\left( \frac{\pi}{6} \right) on the unit circle?
  3. What are some other trigonometric values of common angles (like π/4\pi/4 or π/3\pi/3)?
  4. How does simplifying trigonometric expressions help in solving problems efficiently?
  5. What are the differences between the sine and cosine functions in terms of the unit circle?

Tip: Always double-check the values of standard trigonometric functions for common angles like π/6\pi/6, π/4\pi/4, and π/3\pi/3 so you can solve expressions quickly!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Cosine Function

Formulas

cos(π/6) = √3/2
4 * cos(π/6)

Theorems

Basic Trigonometric Identities

Suitable Grade Level

Grades 9-11