find the five remaining trigonometic functions of θ.\nsecθ = 5/2, sinθ < 0\ncomplete the following…

find the five remaining trigonometic functions of θ.\nsecθ = 5/2, sinθ < 0\ncomplete the following table.\nsinθ = □ cscθ = □\ncosθ = □ secθ = 5/2\ntanθ = □ cotθ = □\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

find the five remaining trigonometic functions of θ.\nsecθ = 5/2, sinθ < 0\ncomplete the following table.\nsinθ = □ cscθ = □\ncosθ = □ secθ = 5/2\ntanθ = □ cotθ = □\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Find (\cos\theta)

Since (\sec\theta=\frac{5}{2}), and (\sec\theta = \frac{1}{\cos\theta}), then (\cos\theta=\frac{2}{5}).

Step2: Determine the quadrant

Given (\sin\theta<0) and (\cos\theta=\frac{2}{5}>0), (\theta) is in the fourth - quadrant.

Step3: Find (\sin\theta)

Using the identity (\sin^{2}\theta+\cos^{2}\theta = 1), substitute (\cos\theta=\frac{2}{5}) into it: (\sin^{2}\theta=1-\cos^{2}\theta=1 - (\frac{2}{5})^{2}=1-\frac{4}{25}=\frac{21}{25}) Since (\sin\theta<0) in the fourth - quadrant, (\sin\theta=-\frac{\sqrt{21}}{5})

Step4: Find (\csc\theta)

Since (\csc\theta=\frac{1}{\sin\theta}), then (\csc\theta=-\frac{5}{\sqrt{21}}=-\frac{5\sqrt{21}}{21})

Step5: Find (\tan\theta)

Using the identity (\tan\theta=\frac{\sin\theta}{\cos\theta}), substitute (\sin\theta =-\frac{\sqrt{21}}{5}) and (\cos\theta=\frac{2}{5}) into it: (\tan\theta=\frac{-\frac{\sqrt{21}}{5}}{\frac{2}{5}}=-\frac{\sqrt{21}}{2})

Step6: Find (\cot\theta)

Since (\cot\theta=\frac{1}{\tan\theta}), then (\cot\theta=-\frac{2}{\sqrt{21}}=-\frac{2\sqrt{21}}{21})

Answer:

(\sin\theta=-\frac{\sqrt{21}}{5}), (\cos\theta=\frac{2}{5}), (\tan\theta=-\frac{\sqrt{21}}{2}), (\csc\theta=-\frac{5\sqrt{21}}{21}), (\cot\theta=-\frac{2\sqrt{21}}{21})