How do you find the value for $sin2theta$ , $cos2theta$ , and $tan2theta$ and the quadrant in which $2theta$ lies given $costheta=-3/5$ and $theta$ is in quadrant III?

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Answer 1 · 2016-09-15T16:31:11

$(1) : sin 2theta=24/25; (2) : cos2theta=-7/25; (3) : tan2theta=-24/7.$

$(4) : 2theta" lies in "Q_(II)$ .

$theta in Q_(III) rArr pi lt theta lt 3pi/2... ... (0) rArr sin theta lt 0$

Now, $cos theta =-3/5 rArr sin theta =-sqrt(1-(-3/5)^2)=-4/5$ .

$:. sin 2theta=2sinthetacostheta=2(-4/5)(-3/5)=24/25... ...(1)$

$cos2theta=2cos^2theta-1=2(-3/5)^2-1=-7/25... ... (2)$

From $(0), 2pi lt theta lt 3pi rArr 2theta in Q_IuuQ_(II).$

But, $(1) & (2)$ confirm that $2theta$ lies in $Q_(II)$ .

Finally, by $(1) & (2), tan 2theta=sin(2theta)/cos(2theta)=-24/7.$

How do you find the value for sin2thetasin2theta, cos2thetacos2theta, and tan2thetatan2theta and the quadrant in which 2theta2theta lies given costheta=-3/5costheta=-3/5 and thetatheta is in quadrant III?