Question

Evaluate exactly without the use of a calculator (remember to rationalize any denominators). a.(cos 60o)(sin 270o) + tan 225o b. – tan 240o + (cos 45o)(sec 135o)

Answer 1

Given:

The trigonometric expressions are given as,

`\begin{gathered} a)\text{ }(\cos 60\degree)(\sin 270\degree)+\tan 225\degree \\ b)-\tan 240\degree+(cos45\degree)(\sec 135\degree) \end{gathered}`

Step-by-step explanation:

a)

The given expression can be rewritten as,

`=(\cos 60\degree)(\sin (360\degree-90\degree))+\tan (270\degree-45\degree)\text{ . . . . .(1)}`

Since, from the trigonometric ratios,

`\begin{gathered} \sin (360\degree-90\degree)=-\sin 90\degree \\ \tan (270\degree-45\degree)=\cot 45\degree \end{gathered}`

On plugging the obtained ratios in equation (1),

`=(\cos 60\degree)(-\sin 90\degree)+\cot 45`

Substitute the trigonometric values in the above equation.

`\begin{gathered} =(1)/(2)(-1)+1 \\ =-(1)/(2)+1 \\ =(1)/(2) \end{gathered}`

Hence, the exact value of the expression is 1/2.

b)

The given expression can be rewritten as,

`=-\tan (270\degree-30\degree)+(\cos 45\degree)(\sec (90\degree+45\degree))\text{ . . . ..(2)}`

Since, from the trigonometric ratios,

`\begin{gathered} \tan (270\degree-30\degree)=\cot 30\degree \\ \sec (90\degree+45\degree)=-\csc 45\degree \end{gathered}`

On plugging the obtained ratios in equation (2),

`=-\cot 30\degree+(\cos 45\degree)(-\csc 45)`

Substitute the trigonometric values in the above equation.

`\begin{gathered} =-\sqrt[]{3}+\frac{1}{\sqrt[]{2}}(-\sqrt[]{2}) \\ =-\sqrt[]{3}-1 \end{gathered}`

Hence, the exact value of the expression is -√3-1.