Question

The terminal side of an angle in standard position passes through the point$\left(-7,-24\right)$.

Calculate the six trigonometric functions for this angle.

Answer 1

Step-1: Determine the distance of the point from the origin:

The terminal side of an angle $\alpha$ passes through the point $\left(-7,-24\right)$.

$\Rightarrow x=-7andy=-24$

The distance of the point from the origin is

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ =\sqrt{49+576} \\ =\sqrt{625} \\ =25 \end{gathered}$$

Step-2: calculating the six trigonometric functions.

$$\begin{gathered} \because \sin \alpha =\frac{y}{r} \\ =\frac{-24}{25} \end{gathered}$$

$$\begin{gathered} \because \cos \alpha =\frac{x}{r} \\ =\frac{-7}{25} \end{gathered}$$

$$\begin{gathered} \because \tan \alpha =\frac{y}{x} \\ =\frac{-24}{-7} \\ =\frac{24}{7} \end{gathered}$$

$$\begin{gathered} \because cot\alpha =\frac{x}{y} \\ =\frac{-7}{-24} \\ =\frac{7}{24} \end{gathered}$$

$$\begin{gathered} \because sec\alpha =\frac{r}{x} \\ =\frac{25}{-7} \\ =\frac{-25}{7} \end{gathered}$$

$$\begin{gathered} \because \cos ec\alpha =\frac{r}{y} \\ =\frac{-25}{24} \end{gathered}$$

Therefore, these are the six trigonometric functions.