cos(pi/12) exact value
Understand the Problem
The question is asking for the exact value of the cosine function at the angle pi/12 radians. To solve this, we can use the cosine of half angles or refer to known angle values in trigonometry.
Answer
$\frac{\sqrt{2 + \sqrt{3}}}{2}$
Answer for screen readers
The exact value of $\cos\left(\frac{\pi}{12}\right)$ is $\frac{\sqrt{2 + \sqrt{3}}}{2}$.
Steps to Solve
-
Identify the angle
We need to find the exact value of $\cos\left(\frac{\pi}{12}\right)$. -
Using the half-angle formula
Recall the half-angle formula for cosine:
$$ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \cos(x)}{2}} $$
We can express $\frac{\pi}{12}$ as a half-angle:
$$ \frac{\pi}{12} = \frac{\pi/6}{2} $$
Where $x = \frac{\pi}{6}$. -
Calculate $\cos\left(\frac{\pi}{6}\right)$
The cosine of $\frac{\pi}{6}$ is known:
$$ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} $$ -
Substitute into the half-angle formula
Now we substitute $\cos\left(\frac{\pi}{6}\right)$ into the half-angle formula:
$$ \cos\left(\frac{\pi}{12}\right) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} $$ -
Simplify the expression
We simplify the expression inside the square root:
First, calculate $1 + \frac{\sqrt{3}}{2}$:
$$ 1 + \frac{\sqrt{3}}{2} = \frac{2}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2} $$
Now substitute it back into the expression:
$$ \cos\left(\frac{\pi}{12}\right) = \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} $$
This simplifies to:
$$ \cos\left(\frac{\pi}{12}\right) = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} $$
The exact value of $\cos\left(\frac{\pi}{12}\right)$ is $\frac{\sqrt{2 + \sqrt{3}}}{2}$.
More Information
The angle $\frac{\pi}{12}$ radians is equivalent to 15 degrees, which is often found in various trigonometric calculations, particularly in solving problems involving regular polygons and angles.
Tips
- A common mistake is to miscalculate the value of cosine for standard angles. Make sure to recall the correct exact values of common angles like $\frac{\pi}{6}$ and $\frac{\pi}{4}$.
- Another mistake is not simplifying the expression correctly after applying the half-angle formula. Always ensure to simplify fully to get the exact form.
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