The terminal side of an angle passes through (-5, 1).
The angle is located at quadrant II, as shown below:
The tangent of the angle is defined as:
[tex]\tan\theta=\frac{y}{x}[/tex]
Calculating:
[tex]\tan\theta=\frac{-5}{1}=-5[/tex]
The cotangent is the reciprocal of the tangent, thus:
[tex]\cot\theta=\frac{1}{-5}=-\frac{1}{5}[/tex]
We can find the secant by using the equation:
[tex]\sec^2\theta=1+\tan^2\theta[/tex]
The secant is negative in quadrant II, so we use the negative root:
[tex]\begin{gathered} \sec^\theta=-\sqrt{1+\tan^2\theta} \\ \sec\theta=-\sqrt{1+(-5)^2} \\ \sec\theta=-\sqrt{26} \end{gathered}[/tex]
The cosine is the reciprocal of the secant, thus:
[tex]\begin{gathered} \cos\theta=\frac{1}{\sec\theta} \\ \cos\theta=-\frac{1}{\sqrt{26}} \end{gathered}[/tex]
The sine can be calculated in several ways. We use, for example:
[tex]\begin{gathered} \sin\theta=\tan\theta\cos\theta \\ \sin\theta=-5\times(-\frac{1}{\sqrt{26}}) \\ \sin\theta=\frac{5}{\sqrt{26}} \end{gathered}[/tex]
Finally, the cosecant is the reciprocal of the sine:
[tex]\begin{gathered} \csc\theta=\frac{1}{\sin\theta} \\ \csc\theta=\frac{1}{5/\sqrt{26}} \\ \csc\theta=\frac{\sqrt{26}}{5} \end{gathered}[/tex]
