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Simplify the trigonometric expression.
Call the expression E:
![\mathsf{E=\dfrac{sin\,\theta}{\sqrt{1-sin^2\,\theta}}\qquad\qquad(but~~1-sin^2\,\theta=cos^2\,\theta)}\\\\\\ \mathsf{E=\dfrac{sin\,\theta}{\sqrt{cos^2\,\theta}}}\\\\\\ \mathsf{E=\dfrac{sin\,\theta}{\left|cos\,\theta\right|}}}](https://tex.z-dn.net/?f=%5Cmathsf%7BE%3D%5Cdfrac%7Bsin%5C%2C%5Ctheta%7D%7B%5Csqrt%7B1-sin%5E2%5C%2C%5Ctheta%7D%7D%5Cqquad%5Cqquad%28but~~1-sin%5E2%5C%2C%5Ctheta%3Dcos%5E2%5C%2C%5Ctheta%29%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7BE%3D%5Cdfrac%7Bsin%5C%2C%5Ctheta%7D%7B%5Csqrt%7Bcos%5E2%5C%2C%5Ctheta%7D%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7BE%3D%5Cdfrac%7Bsin%5C%2C%5Ctheta%7D%7B%5Cleft%7Ccos%5C%2C%5Ctheta%5Cright%7C%7D%7D%7D)
![\mathsf{E}=\left\{\!\begin{array}{rl}\mathsf{\dfrac{sin\,\theta}{cos\,\theta}\,,}&\mathsf{\quad if~~cos\,\theta>0}\\\\\mathsf{\dfrac{sin\,\theta}{-\,cos\,\theta}\,,}&\mathsf{\quad if~~cos\,\theta](https://tex.z-dn.net/?f=%5Cmathsf%7BE%7D%3D%5Cleft%5C%7B%5C%21%5Cbegin%7Barray%7D%7Brl%7D%5Cmathsf%7B%5Cdfrac%7Bsin%5C%2C%5Ctheta%7D%7Bcos%5C%2C%5Ctheta%7D%5C%2C%2C%7D%26%5Cmathsf%7B%5Cquad%20if~~cos%5C%2C%5Ctheta%3E0%7D%5C%5C%5C%5C%5Cmathsf%7B%5Cdfrac%7Bsin%5C%2C%5Ctheta%7D%7B-%5C%2Ccos%5C%2C%5Ctheta%7D%5C%2C%2C%7D%26%5Cmathsf%7B%5Cquad%20if~~cos%5C%2C%5Ctheta%3C0%7D%5Cend%7Barray%7D%5Cright.%5C%5C%5C%5C%5C%5C%5C%5C%5Cmathsf%7BE%7D%3D%5Cleft%5C%7B%5C%21%5Cbegin%7Barray%7D%7Brr%7D%5Cmathsf%7Btan%5C%2C%5Ctheta%2C%5Cqquad%20if%7D%26%5Cmathsf%7B-%5Cdfrac%7B%5Cpi%7D%7B2%7D%2Bk%5Ccdot%202%5Cpi%3C%5Ctheta%3C%5Cdfrac%7B%5Cpi%7D%7B2%7D%2Bk%5Ccdot%202%5Cpi%7D%5C%5C%5C%5C%5Cmathsf%7B-%5C%2Ctan%5C%2C%5Ctheta%2C%5Cqquad%20if%7D%26%5Cmathsf%7B%5Cdfrac%7B%5Cpi%7D%7B2%7D%2Bk%5Ccdot%202%5Cpi%3C%5Ctheta%3C%5Cdfrac%7B3%5Cpi%7D%7B2%7D%2Bk%5Ccdot%202%5Cpi%7D%5Cend%7Barray%7D%5Cright.%5Cqquad%5Cmathsf%7Bk%20%5Cin%20%5Cmathbb%7BZ%7D%7D)
So basically E is equivalent to
![\pm\,\mathsf{tan\,\theta},](https://tex.z-dn.net/?f=%5Cpm%5C%2C%5Cmathsf%7Btan%5C%2C%5Ctheta%7D%2C)
where the sign depends on which quadrant
![\theta](https://tex.z-dn.net/?f=%5Ctheta)
lies. Shortly,
![\mathsf{\dfrac{sin\,\theta}{\sqrt{1-sin^2\,\theta}}}=\left\{\! \begin{array}{rl} \mathsf{tan\,\theta,}&\mathsf{\quad if~\theta~lies~either~in~the~1^{st}~or~the~4^{th}~quadrant}\\\\ \mathsf{-\,tan\,\theta,}&\mathsf{\quad if~\theta~lies~either~in~the~2^{nd}~or~the~3^{rd}~quadrant} \end{array}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bsin%5C%2C%5Ctheta%7D%7B%5Csqrt%7B1-sin%5E2%5C%2C%5Ctheta%7D%7D%7D%3D%5Cleft%5C%7B%5C%21%0A%5Cbegin%7Barray%7D%7Brl%7D%0A%5Cmathsf%7Btan%5C%2C%5Ctheta%2C%7D%26%5Cmathsf%7B%5Cquad%20if~%5Ctheta~lies~either~in~the~1%5E%7Bst%7D~or~the~4%5E%7Bth%7D~quadrant%7D%5C%5C%5C%5C%0A%5Cmathsf%7B-%5C%2Ctan%5C%2C%5Ctheta%2C%7D%26%5Cmathsf%7B%5Cquad%20if~%5Ctheta~lies~either~in~the~2%5E%7Bnd%7D~or~the~3%5E%7Brd%7D~quadrant%7D%0A%5Cend%7Barray%7D)
I hope this helps. =)
Tags: <em>simplify trigonometric trig expression sine cosine tangent sin cos tan absolute value modulus trigonometry</em>
It would look something like the image attached.
Answer:
The answer is neither.
Step-by-step explanation:
The slope of AB is -5/0, and the slope of CD is 4/0.
Answer:
0.45 45%
Step-by-step explanation:
Answer:
the answer is (2,-6)
Step-by-step explanation: