What is the exact value of cos(π/12)?
Understand the Problem
The question is asking for the exact value of the cosine of the angle π/12 radians. To solve this, we can use the half-angle or angle subtraction formulas for cosine since π/12 can be expressed in terms of known angles.
Answer
The exact value is $\frac{\sqrt{2} + \sqrt{6}}{4}$.
Answer for screen readers
The exact value of $\cos\left(\frac{\pi}{12}\right)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$.
Steps to Solve
- Identify angles to use in the cosine formula
We can express $\frac{\pi}{12}$ as a difference of angles. Notably, we’ll use $\frac{\pi}{3}$ (which is $60^\circ$) and $\frac{\pi}{4}$ (which is $45^\circ$):
$$ \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} $$
- Apply the cosine angle subtraction formula
The cosine angle subtraction formula states:
$$ \cos(a - b) = \cos a \cos b + \sin a \sin b $$
Substituting $a = \frac{\pi}{3}$ and $b = \frac{\pi}{4}$ into the formula gives:
$$ \cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{4}\right) $$
- Calculate the cosine and sine values
Now we need to find the cosine and sine of the angles:
- $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$
- $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
- $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
- $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
- Substitute the values back into the formula
Now, substituting the values we found in the previous step, we have:
$$ \cos\left(\frac{\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$
- Simplify the expression
Now simplify each term:
$$ \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} $$
Combine the terms:
$$ \cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2} + \sqrt{6}}{4} $$
The exact value of $\cos\left(\frac{\pi}{12}\right)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$.
More Information
Using trigonometric identities like the cosine angle subtraction formula allows us to find exact values of trigonometric functions for non-standard angles. The value $\frac{\pi}{12}$ is useful in various applications in mathematics, especially in geometry and calculus.
Tips
- Mistaking the angle subtraction formula by incorrectly placing the sine and cosine functions.
- Forgetting to use the values of sine and cosine for angles like $\frac{\pi}{3}$ and $\frac{\pi}{4}$ correctly.
- Not simplifying fractions properly.
AI-generated content may contain errors. Please verify critical information
