Given:
Given θ is an angle in Quadrant III.
[tex]\sin\theta=-\frac{3}{5}[/tex]Required:
To find the value of secθ and cotθ.
Explanation:
In Quadrant III, cosine (the x-value of the unit circle) and sine (the y-value of the unit circle) are both negative and tangent is positive.
[tex]\begin{gathered} \sin\theta=-\frac{3}{5} \\ \\ =-\frac{opp}{adj} \end{gathered}[/tex]To find the adjacent side of the triangle use the Pythagorean Theorem:
[tex]\begin{gathered} x^2+(-3)^2=5^2 \\ x^2=25-9 \\ x^2=16 \\ x=\pm4 \end{gathered}[/tex]Since we are in the third quadrant
[tex]x=-4[/tex]One way to find the secant and cotangent is to use the inverse identities:
[tex]\begin{gathered} sec\theta=\frac{1}{\cos\theta} \\ \\ =\frac{1}{-\frac{4}{5}} \\ \\ =-\frac{5}{4} \end{gathered}[/tex]And
[tex]\begin{gathered} cot\theta=\frac{1}{\tan\theta} \\ \\ =\frac{1}{-\frac{3}{-4}} \\ \\ =\frac{4}{3} \end{gathered}[/tex]Final Answer:
[tex]\begin{gathered} sec\theta=-\frac{5}{4} \\ \\ cot\theta=\frac{4}{3} \end{gathered}[/tex]