Question

Which of the following has the same value as cos 3pi

Answer 1

Option C :
`\sin \left(-(\pi)/(2)\right)` has the same value as
`\cos (3 \pi)`

Step-by-step explanation:

It is given that to find the same value as
`\cos (3 \pi)` from the 4 options.

Now, we shall find the value of
`\cos (3 \pi)`

`\cos (3 \pi)=\cos (2 \pi+\pi)`

`\cos (3 \pi)=\cos (\pi)`

The value of
`\cos (\pi)` is -1.

Thus, the value of
`\cos (3 \pi)` is -1.

Option A :
`\sec \left((3 \pi)/(2)\right)`

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

Using the identity
`\cos (x)=\sin \left((\pi)/(2)-x\right)`,

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

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

`\sec \left((3 \pi)/(2)\right)=\frac{1}{\sin \left(-{ \pi}\right)}`

Since, the value of
`\sin (\pi)` is 0. Thus, we have,

`\sec \left((3 \pi)/(2)\right)=(1)/(0\right))`

Hence, the value of
`\sec \left((3 \pi)/(2)\right)` is undefined.

Thus, the value of
`\sec \left((3 \pi)/(2)\right)` is not the same as value of
`\cos (3 \pi)`

Therefore, Option A is not the correct answer.

Option B :
`\tan \left((\pi)/(2)\right)`

The value of
`\tan \left((\pi)/(2)\right)` is undefined.

Thus, the value of
`\tan \left((\pi)/(2)\right)` is not the same as value of
`\cos (3 \pi)`

Therefore, Option B is not the correct answer.

Option C :
`\sin \left((-\pi)/(2)\right)`

`\sin \left(-(\pi)/(2)\right)=-\sin \left((\pi)/(2)\right)`

Since,
`\sin \left((\pi)/(2)\right)=1` Substituting, we have,

`\sin \left((-\pi)/(2)\right)=-1`

Thus, the value of
`\sin \left((-\pi)/(2)\right)` is the same as the value of
`\cos (3 \pi)`

Therefore, Option C is the correct answer.

Option D :
`\cot (-\pi)`

Rewriting the angles,

`\cot (-\pi)=\cot (-\pi+0\pi)`

Simplifying, we get,

`\cot (-\pi)=\cot (0 \pi)`

The value of
`\cot (0 \pi)` is undefined.

Thus, the value of
`\cot (-\pi)` is not the same as the value of
`\cos (3 \pi)`

Therefore, Option D is not the correct answer.