Question

find the five remaining trigonometic functions of θ.
secθ = 5/2, sinθ < 0
complete the following table.
sinθ = □ cscθ = □
cosθ = □ secθ = 5/2
tanθ = □ cotθ = □
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

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:

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