Question 1

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

$give the exact answers do not use decimal numbers the answer should be a fraction-example-1$

Question 2

Solution

We are given

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

To find cos(2t)

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

To find sin(2t)

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

To find cos(t/2)

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

To find sin(t/2)

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