Find sin⁡θ if cot⁡θ=−2 and cos⁡θ&lt;0.

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Find sin⁡θ if cot⁡θ=−2 and cos⁡θ&amp;lt;0.

braodagxj · Accepted Answer

Step 1&amp;nbsp;To determine the value of&amp;nbsp;sin⁡(θ).&amp;nbsp;Step 2&amp;nbsp;Given information:&amp;nbsp;The following table lists the trigonometric functions.&amp;nbsp;cot⁡(θ)=−2&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;cos⁡(θ)&amp;lt;0&amp;nbsp;Step 3&amp;nbsp;Used concept:&amp;nbsp;Pythagoras theorem:&amp;nbsp;If the base is x, the hypotenuse is z, and y is the perpendicular,Step 4&amp;nbsp;Calculation:&amp;nbsp;cot⁡(θ)&amp;nbsp;is given below.&amp;nbsp;cot⁡(θ)=baseperpendica―r&amp;nbsp;=xy&amp;nbsp;=−21&amp;nbsp;so,&amp;nbsp;x=−2&amp;nbsp;&amp;nbsp;and&amp;nbsp;&amp;nbsp;y=1&amp;nbsp;Step 5&amp;nbsp;From the Pythagoras theorem.&amp;nbsp;z2=x2+y2&amp;nbsp;=(−2)2+(1)2&amp;nbsp;=4+1&amp;nbsp;z2=5&amp;nbsp;z=±5&amp;nbsp;Step 6&amp;nbsp;Since&amp;nbsp;cos⁡(θ)&amp;lt;0, so take the value of z with positive sign. Then&amp;nbsp;The value of&amp;nbsp;sin⁡(θ)&amp;nbsp;is given below.&amp;nbsp;sin⁡(θ)=perpendica―rhypoteνse&amp;nbsp;=yz&amp;nbsp;=15&amp;nbsp;sin⁡(θ)=15

Ronnie Schechter · Accepted Answer

Start with the identity:  1+cot2(θ)=csc2(θ)  Explanation:   Substitute cot2(θ)=(−2)2:  1+(−2)2=csc2(θ)  5=csc2(θ)  Substitute csc2(θ)=1sin2(θ)  5=1sin2(θ)  sin2(θ)=15  sin⁡(θ)=±55  Because we are told that cos⁡(θ)&amp;lt;0 and cot⁡(θ)=−2, we know that the sine function must be positive in this quadrant:  sin⁡(θ)=55

Jeffrey Jordon · Accepted Answer

Given was&amp;nbsp;cot⁡θ=−21&amp;nbsp;and cot is the reciprocal of tanthereforetan⁡θ=1−2Since the adjacent sides b=-2 and the opposite side a=1 are already known, find the hypotenuse c. That is,c2=a2+b2c=12+(−2)2c=5Solving for&amp;nbsp;sin⁡θsin⁡θ&amp;nbsp;=oppositesides*hypotenuse=absin⁡θ=15&amp;nbsp;need to rationalizesin⁡θ=(15)×55sin⁡θ=55Note that the adjacent side of the angle is -2 so the angle&amp;nbsp;θ&amp;nbsp;is found in the 2nd quadrant.

Find \sin \theta\ if\ \cot \theta = - 2\ and\

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