sec⁡θ=−3,tan⁡θ&gt;0. Find the exact value of the remaining trigonometric functions of θ.

Question

sec⁡θ=−3,tan⁡θ&amp;gt;0. Find the exact value of the remaining trigonometric functions of θ.

bahaistag · Accepted Answer

We have given the value of a trigonometric function sec⁡(θ) and asked the values of other remaining trigonometric functions. So here we will use some standard identities and results in order to determine the values of other trigonometric functions.  sec2θ−tan2θ=1  cos⁡θ=1sec⁡θ  sin⁡θ=tan⁡θsec⁡θ  cos⁡ecθ=1sin⁡θ  cot⁡θ=1tan⁡θ  Since, sec⁡θ=−3&amp;lt;0andtan⁡θ&amp;gt;0 then θ must be in third quadrant in which sin⁡θ,cos⁡θ,cos⁡ecθandsec⁡θ will be negative and rest tan⁡θandcot⁡θ will be positive.  Now,   sec2θ−tan2θ=1  (−3)2−tan2θ=1  tan2θ=(−3)2−1  tan⁡θ=8  Also,  cos⁡θ=1sec⁡θ=1−3=−13  sin⁡θ=tan⁡θsec⁡θ=8−3=−223  cos⁡ecθ=1sin⁡θ=1−223=−322  cot⁡θ=1tan⁡θ=18=122  Hence, values of other trigonometric functions sin⁡θ,cos⁡θ,tan⁡θcos⁡ecθandcot⁡θ will be −223,−13,22,−322and122 respectively.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

sec theta = -3, tan theta > 0. Find the exact value of the remaining trigonometric functions of theta.

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