Answer:
[tex] \displaystyle \text C)\csc( {225}^{ \circ } )= - \sqrt{2} [/tex]
Step-by-step explanation:
we want to figure out the exact value of the following:
[tex] \displaystyle \csc( {225}^{ \circ} ) [/tex]
recall that,
[tex] \displaystyle \csc( \theta ) = \frac{1}{ \sin( \theta)} [/tex]
we are given [tex]\theta=225°[/tex]
substitute:
[tex] \displaystyle \csc( {225}^{ \circ } )= \frac{1}{ \sin( {225}^{ \circ} )} [/tex]
225° belongs to Q:III which we can clearly acquire from unit circle so,
[tex] \displaystyle \csc( {225}^{ \circ } )= \frac{1}{ \dfrac{ - \sqrt{2} }{2} } [/tex]
By simplifying complex fraction we get:
[tex] \displaystyle \csc( {225}^{ \circ } )= \frac{2}{ - \sqrt{ 2} } [/tex]
notice that, what we acquired isn't in the option so multiply it by √2/√2
[tex] \displaystyle \csc( {225}^{ \circ } )= \frac{2}{ - \sqrt{ 2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } [/tex]
simplify multiplication:
[tex] \displaystyle \csc( {225}^{ \circ } )= \frac{2 \sqrt{2} }{ - \sqrt{ 4} } [/tex]
simplify square:
[tex] \displaystyle \csc( {225}^{ \circ } )= \frac{2 \sqrt{2} }{ - 2} [/tex]
reduce fraction:
[tex] \displaystyle \csc( {225}^{ \circ } )= - \sqrt{2} [/tex]
hence,
our answer is C
