Question

Type the correct answer in each box.

If cos -105º = -0.26 and csc -105º = -1.03, then cos 105º = ? and csc 105º = ?

Answer 1

`\cos(105^\circ)\approx-0.26\text{ and } \csc(105^\circ)\approx1.03`

Explanation:

We are given that:

`\cos(-105^\circ)\approx-0.26\text{ and } \csc(-105^\circ)\approx-1.03`

And we want to find:

`\cos(105^\circ)\text{ and } \csc(105^\circ)`

Part A)

Remember that cosine (and secant) is an even function. By definition, this means that:

`\cos(\theta^\circ)=\cos(-\theta^\circ)`

Therefore, since we know that cos(-105°) is about -0.26, it follows from the definition that:

`\cos(-105^\circ)=\cos(105^\circ)\approx-0.26`

Part B)

Remember that cosecant (and sine) is an odd function. By definition, this means that:

`\csc(-\theta^\circ)=-\csc(\theta^\circ)`

We know that csc(-105°) is about -1.03.

By the above definition, we can rewrite csc(105°) as csc(-(-105°)) or -csc(-105°).

Hence, it follows that:

`\csc(105^\circ)=-\csc(-105^\circ)=-(-1.03)=1.03`