Determine the value of $\cos(\frac{7\pi}{12})$.
Understand the Problem
The question asks to find the value of $\cos(\frac{7\pi}{12})$. This requires using trigonometric identities to express $\frac{7\pi}{12}$ as a sum or difference of angles whose cosine and sine values are known, such as $\frac{\pi}{3}$, $\frac{\pi}{4}$, and $\frac{\pi}{6}$. We can use the cosine addition formula to evaluate it. $\frac{7\pi}{12}$ can be expressed as $\frac{\pi}{3} + \frac{\pi}{4}$.
Answer
$\frac{\sqrt{2} - \sqrt{6}}{4}$
Answer for screen readers
$\frac{\sqrt{2} - \sqrt{6}}{4}$
Steps to Solve
- Express $\frac{7\pi}{12}$ as a sum of known angles
We can express $\frac{7\pi}{12}$ as a sum of $\frac{\pi}{3}$ and $\frac{\pi}{4}$:
$$ \frac{7\pi}{12} = \frac{4\pi}{12} + \frac{3\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} $$
- Apply the cosine addition formula
The cosine addition formula is: $$ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) $$ In our case, $a = \frac{\pi}{3}$ and $b = \frac{\pi}{4}$. So we have:
$$ \cos\left(\frac{7\pi}{12}\right) = \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right) $$
- Evaluate the trigonometric functions
We know that: $$ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} $$ $$ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$ $$ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} $$ $$ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$
- Substitute the values and simplify
Substitute the known values into the equation:
$$ \cos\left(\frac{7\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$ $$ \cos\left(\frac{7\pi}{12}\right) = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} $$ $$ \cos\left(\frac{7\pi}{12}\right) = \frac{\sqrt{2} - \sqrt{6}}{4} $$
$\frac{\sqrt{2} - \sqrt{6}}{4}$
More Information
The value of $\cos(\frac{7\pi}{12})$ is a negative number, approximately -0.2588. This is because $\frac{7\pi}{12}$ is in the second quadrant, where cosine values are negative.
Tips
A common mistake is to incorrectly remember the cosine addition formula, especially the sign between the two terms. Another mistake can occur when evaluating sine and cosine of $\frac{\pi}{3}$ and $\frac{\pi}{4}$. Finally, mistakes can occur when simplifying the expression at the end.
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