Evaluate the expression:
$\sin \left(\frac{\pi}{3}\right)+\cos \left(\frac{\pi}{3}\right)$
To evaluate the expression \(\sin \left(\frac{\pi}{3}\right) + \cos \left(\frac{\pi}{3}\right)\), we start by identifying the trigonometric values for the angle \(\frac{\pi}{3}\) radians (which is equivalent to \(60^\circ\)). These are fundamental values found on the unit circle.
Using standard trigonometric definitions, we determine the values for the sine and cosine of \(\frac{\pi}{3}\):
\[\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\]
\[\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}\]
Now, we substitute these values into the given expression:
\[\sin \left(\frac{\pi}{3}\right) + \cos \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} + \frac{1}{2}\]
Combining these fractions with a common denominator, we get:
\[\frac{\sqrt{3} + 1}{2}\]
Thus, the value of the expression is \(\frac{\sqrt{3} + 1}{2}\).
Trigonometric values like these are essential in physics and engineering, particularly when resolving forces or vectors into components. For instance, if a force is acting at an angle of \(60^\circ\) (\(\frac{\pi}{3}\)) to the horizontal, the horizontal and vertical components of the force vector are determined by the cosine and sine of that angle. Understanding how to sum these components correctly is vital for calculating the resultant forces in structures like bridges or mechanical assemblies.