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# find sin theta

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If cos(θ) = − 10/11 with θ in Quadrant III, what is sin(θ)?

Oct 10, 2017

#1
+7352
+2

From the Pythagorean identity,

sin2 θ  +  cos2 θ     =  1        Plug in  -10/11  for  cos θ .

sin2 θ  +  (-10/11)2  =  1       $$(-\frac{10}{11})^2=(-\frac{10}{11})(-\frac{10}{11})=\frac{100}{121}$$

sin2 θ  +  100/121   =  1       Subtract   100 / 121  from both sides.

sin2 θ   =   21/121                Since sin is negative in Quad III, take the negative sqrt of both sides.

sin θ     =  $$-\sqrt{\frac{21}{121}}$$

sin θ     =   $$-\frac{\sqrt{21}}{11}$$

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Oct 10, 2017
edited by hectictar  Oct 10, 2017

#1
+7352
+2

From the Pythagorean identity,

sin2 θ  +  cos2 θ     =  1        Plug in  -10/11  for  cos θ .

sin2 θ  +  (-10/11)2  =  1       $$(-\frac{10}{11})^2=(-\frac{10}{11})(-\frac{10}{11})=\frac{100}{121}$$

sin2 θ  +  100/121   =  1       Subtract   100 / 121  from both sides.

sin2 θ   =   21/121                Since sin is negative in Quad III, take the negative sqrt of both sides.

sin θ     =  $$-\sqrt{\frac{21}{121}}$$

sin θ     =   $$-\frac{\sqrt{21}}{11}$$

hectictar Oct 10, 2017
edited by hectictar  Oct 10, 2017