Given:
Given that a trigonometric statement.
Required:
To check whether the given statements are true or false`.
Explanation:
(1)
Now consider LHS side of the given statement,
[tex]\cos(-\frac{5\pi}{3})=\cos(2\pi-\frac{5\pi}{3})[/tex]Because adding 2π does not change the value of a trigonometric function.
Now,
[tex]\begin{gathered} \cos(2\pi-\frac{5\pi}{3})=\cos\frac{\pi}{3} \\ \\ =\frac{1}{2} \end{gathered}[/tex]Therefore, the statement is false.
(2)
[tex]\begin{gathered} cot(-\frac{11\pi}{3})=cot\frac{\pi}{3} \\ \\ =\frac{1}{\sqrt{3}} \end{gathered}[/tex]Therefore, the statement is False.
(3)
Remove full rotations of 2π until the angle is between 0 and 2π.
[tex]\begin{gathered} \sin(\frac{27\pi}{2})=\sin(\frac{3\pi}{2}) \\ \\ =-1 \end{gathered}[/tex]Therefore, the statement is TRUE.
Final Answer:
The first and second statement are FALSE and the third one is TRUE.
.