Sin PI/5 cos 3PI/10 + cos PI/5 sin 3PI/10?
The Correct Answer and Explanation is :
he expression ( \sin\left(\frac{\pi}{5}\right) \cos\left(\frac{3\pi}{10}\right) + \cos\left(\frac{\pi}{5}\right) \sin\left(\frac{3\pi}{10}\right) ) can be simplified using the sine addition formula, which states:
[ \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) ]
y comparing this formula to the given expression, we can identify ( A = \frac{\pi}{5} ) and ( B = \frac{3\pi}{10} ). Applying the formula:
[ \sin\left(\frac{\pi}{5} + \frac{3\pi}{10}\right) ]
o combine the angles, we need a common denominator. Converting both fractions to have a denominator of 10:
[ \frac{\pi}{5} = \frac{2\pi}{10} ]
dding the angles:
[ \frac{2\pi}{10} + \frac{3\pi}{10} = \frac{5\pi}{10} = \frac{\pi}{2} ]
herefore:
[ \sin\left(\frac{\pi}{2}\right) ]
he sine of ( \frac{\pi}{2} ) is 1:
[ \sin\left(\frac{\pi}{2}\right) = 1 ]
hus, the value of the original expression is 1.
Explanation:
rigonometric identities are powerful tools that allow us to simplify and evaluate expressions involving trigonometric functions.he sine addition formula, in particular, expresses the sine of a sum of two angles in terms of the sines and cosines of the individual angles:
[ \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) ]
his identity is derived from the unit circle and the definitions of the sine and cosine functions.t is especially useful when dealing with expressions that match its structure, as it allows for the combination of terms into a single trigonometric function.
n the given problem, the expression ( \sin\left(\frac{\pi}{5}\right) \cos\left(\frac{3\pi}{10}\right) + \cos\left(\frac{\pi}{5}\right) \sin\left(\frac{3\pi}{10}\right) ) directly matches the right-hand side of the sine addition formula, with ( A = \frac{\pi}{5} ) and ( B = \frac{3\pi}{10} ).y recognizing this pattern, we can simplify the expression to ( \sin\left(\frac{\pi}{5} + \frac{3\pi}{10}\right) ).
ombining the angles requires finding a common denominator.onverting ( \frac{\pi}{5} ) to ( \frac{2\pi}{10} ) allows us to add it to ( \frac{3\pi}{10} ), resulting in ( \frac{5\pi}{10} ), which simplifies to ( \frac{\pi}{2} ).he sine of ( \frac{\pi}{2} ) is a fundamental value in trigonometry, known to be 1.
his process highlights the elegance of trigonometric identities in simplifying complex expressions.y applying the sine addition formula, we reduced the original expression to a basic trigonometric value, demonstrating the utility of these identities in mathematical problem-solving.