use the fact that the trigonometric functions are periodic to find the exact value of the given…

use the fact that the trigonometric functions are periodic to find the exact value of the given expressior\n\n\\( \\sin \\frac { 13 \\pi } { 3 } \\)\n\n\\( \\sin \\frac { 13 \\pi } { 3 } = \\square \\)\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressi

use the fact that the trigonometric functions are periodic to find the exact value of the given expressior\n\n\\( \\sin \\frac { 13 \\pi } { 3 } \\)\n\n\\( \\sin \\frac { 13 \\pi } { 3 } = \\square \\)\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressi

Answer

Explanation:

Step1: Find the period of the sine function

The period of the sine function (y = \sin(x)) is (2\pi). We can write (\frac{13\pi}{3}=4\pi+\frac{\pi}{3}). Since (\sin(x + 2k\pi)=\sin(x)) for any integer (k), and here (k = 2) (because (4\pi=2\times2\pi)), we have (\sin(\frac{13\pi}{3})=\sin(4\pi+\frac{\pi}{3})).

Step2: Simplify using the periodicity

By the periodicity property (\sin(x + 2k\pi)=\sin(x)), when (x=\frac{\pi}{3}) and (k = 2), we get (\sin(4\pi+\frac{\pi}{3})=\sin(\frac{\pi}{3})).

Step3: Evaluate (\sin(\frac{\pi}{3}))

We know that (\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}) from the unit - circle definition of the sine function (for a (30 - 60-90) triangle with hypotenuse (r = 1), opposite side (y=\frac{\sqrt{3}}{2}) when the angle is (\frac{\pi}{3})).

Answer:

(\frac{\sqrt{3}}{2})