Question

give the exact answers do not use decimal numbers the answer should be a fraction or an atithmetic expression

Answer 1

Solution

We are given

$cost=-\frac{3}{4}\text{ on }\piSo we are on the third quadrant and only tangent is positive at the third quadrant[tex]\begin{gathered} sin^2t+cos^2t=1 \\ \\ sin^2t=1-cos^2t \\ \\ sin^2t=1-(-\frac{3}{4})^2 \\ \\ sin^2t=1-\frac{9}{16} \\ \\ sin^2t=\frac{7}{16} \\ \\ sint=-\frac{\sqrt{7}}{4}\text{ \lparen since we are on the third quadrant\rparen} \end{gathered}$

To find cos(2t)

$\begin{gathered} cos2t=2cos^2t-1 \\ \\ cos2t=2(-\frac{3}{4})^2-1 \\ \\ cos2t=2(\frac{9}{16})-1 \\ \\ cos2t=\frac{9}{8}-1 \\ \\ cos2t=\frac{1}{8} \end{gathered}$

To find sin(2t)

$\begin{gathered} sin2t=2sintcost \\ \\ sin2t=2(-\frac{\sqrt{7}}{4})(-\frac{3}{4}) \\ \\ sin2t=\frac{3\sqrt{7}}{8} \end{gathered}$

To find cos(t/2)

$\begin{gathered} cos(t)=2cos^2(\frac{t}{2})-1 \\ \\ cos(t)+1=2cos^2(\frac{t}{2}) \\ \\ -\frac{3}{4}+1=2cos^2(\frac{t}{2}) \\ \\ \frac{1}{4}=2cos^2(\frac{t}{2}) \\ \\ \frac{1}{8}=cos^2(\frac{t}{2}) \\ \\ cos^2(\frac{t}{2})=\frac{1}{8} \\ \\ cos(\frac{t}{2})=-\frac{1}{2\sqrt{2}} \\ \\ cos(\frac{t}{2})=-\frac{\sqrt{2}}{4} \end{gathered}$

To find sin(t/2)

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