Question

Show with work please.

Answer 1

`$\csc \left(\theta-(\pi )/(2)\right)=0.73$`

Explanation:

The identity you will use is:

`$\csc \left(x\right)=(1)/(\sin \left(x\right))$`

So,

`$\csc \left(\theta-(\pi )/(2)\right)$`

`$\csc \left(\theta-(\pi )/(2)\right)=(1)/(\sin \left(-(\pi )/(2)+\theta\right))$`

Now, using the difference of sin

Note: state that
`\text{sin}(\alpha\pm \beta)=\text{sin}(\alpha) \text{cos}(\beta) \pm \text{cos}(\alpha) \text{sin}(\beta)`

`$\csc \left(\theta-(\pi )/(2)\right)=(1)/(-\cos \left(\theta\right)\sin \left((\pi )/(2)\right)+\cos \left((\pi )/(2)\right)\sin \left(\theta\right))$`

Solving the difference of sin:

`$-\cos \left(\theta\right)\sin \left((\pi )/(2)\right)+\cos \left((\pi )/(2)\right)\sin \left(\theta\right)$`

`-\cos \left(\theta\right) \cdot 1+0\cdot \sin \left(\theta\right)`

`-\text{cos} \left(\theta\right)`

Then,

`$\csc \left(\theta-(\pi )/(2)\right)=-(1)/(\cos \left(\theta\right))$`

Once

`\text{sec}(-\theta)=\text{sec}(\theta)`

And,
`\text{sec}(\theta)=-0.73`

`$-(1)/(\cos \left(\theta\right))=-\text{sec}(\theta)$`

`$-(1)/(\cos \left(\theta\right))=-(-0.73)$`

`$-(1)/(\cos \left(\theta\right))=0.73$`

Therefore,

`$\csc \left(\theta-(\pi )/(2)\right)=0.73$`