Question

What is the exact vale of a trigonometric function of a given angle?

Select True or False for each statement .
sin (5π/6) = - 1/2
tan (27π) = 0
sec (- 11π/3) = - 2√3/3​

Answer 1

1) False

2) True

3) False

Explanation:

From inspection of the attached unit circle, we can see that:

`\boxed{\sin\left((5\pi)/(6)\right)=(1)/(2)}`

Therefore, the first statement is false.

`\hrulefill`

The tangent function is periodic with a period of π, meaning that for any integer n:

`\tan(a)=\tan(a+\pi n)`

`\tan(n\pi)=\tan \pi`

Therefore:

`\tan(27\pi)=\tan \pi`

From the unit circle we can see that:

`\tan \pi=(\sin \pi)/(\cos \pi)=(0)/(-1)=0`

Therefore,

`\boxed{\tan(27\pi)=0}`

Therefore, the second statement is true.

`\hrulefill`

`\textsf{Using the trigonometric identity\; $\sec x=(1)/(\cos x)$}:`

`\sec \left(-(11\pi)/(3)\right)=(1)/(\cos \left(-(11\pi)/(3)\right))`

The cosine function is periodic with a period of 2π, meaning that for any integer n:

`\cos (a)=\cos (a + 2\pi n)`

Therefore:

`\cos \left(-(11\pi)/(3)\right)=\cos \left(-(11\pi)/(3)+4\pi\right)=\cos \left((\pi)/(3)\right)`

From the unit circle we can see that:

`\cos \left((\pi)/(3)\right)=(1)/(2)`

Therefore:

`\sec \left(-(11\pi)/(3)\right)=(1)/(\cos \left(-(11\pi)/(3)\right))=(1)/(\cos \left((\pi)/(3)\right))=(1)/((1)/(2))=2`

Therefore, the third statement is false.