Math / Trigonometry

Question $\sin \frac{\pi}{3} \cos \frac{\pi}{6}+\cos \frac{\pi}{3} \sin \frac{\pi}{6}$

Studdy Solution

STEP 1

1. The problem involves trigonometric functions and specific angles.
2. We can use trigonometric identities to simplify the expression.
3. The angles given are standard angles with known sine and cosine values.

STEP 2

1. Identify and apply the appropriate trigonometric identity.
2. Calculate the sine and cosine values for the given angles.
3. Simplify the expression using the calculated values.

STEP 3

Recognize that the expression $\sin \frac{\pi}{3} \cos \frac{\pi}{6} + \cos \frac{\pi}{3} \sin \frac{\pi}{6}$ matches the form of the sine addition formula:
$\sin A \cos B + \cos A \sin B = \sin(A + B)$
In this case, $A = \frac{\pi}{3}$ and $B = \frac{\pi}{6}$ .
Apply the identity:
$\sin\left(\frac{\pi}{3} + \frac{\pi}{6}\right)$

STEP 4

Calculate the sum of the angles:
$\frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}$

STEP 5

Find the sine of the resulting angle:
$\sin\left(\frac{\pi}{2}\right) = 1$
Thus, the value of the original expression is:
$\boxed{1}$

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