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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​

What is the exact vale of a trigonometric function of a given angle? Select True or-example-1

1 Answer

4 votes

Answer:

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.

What is the exact vale of a trigonometric function of a given angle? Select True or-example-1
User Chhorn Elit
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