To find the values of [tex]\(\cos 105^\circ\)[/tex] and [tex]\(\csc 105^\circ\)[/tex] given the values of [tex]\(\cos -105^\circ = -0.26\)[/tex] and [tex]\(\csc -105^\circ = -1.03\)[/tex], we can use the properties of trigonometric functions for negative angles. These properties are:
1. [tex]\(\cos(-\theta) = \cos(\theta)\)[/tex]
2. [tex]\(\csc(-\theta) = -\csc(\theta)\)[/tex]
Given:
[tex]\[
\cos(-105^\circ) = -0.26
\][/tex]
Using the identity for cosine of a negative angle:
[tex]\[
\cos 105^\circ = \cos(-105^\circ)
\][/tex]
Therefore,
[tex]\[
\cos 105^\circ = -0.26
\][/tex]
Next, given:
[tex]\[
\csc(-105^\circ) = -1.03
\][/tex]
Using the identity for cosecant of a negative angle:
[tex]\[
\csc 105^\circ = -\csc(-105^\circ)
\][/tex]
Therefore,
[tex]\[
\csc 105^\circ = 1.03
\][/tex]
So, we have:
[tex]\[
\cos 105^\circ = -0.26
\][/tex]
and
[tex]\[
\csc 105^\circ = 1.03
\][/tex]
