Question

If Ois an angle in standard position and its terminal side passes through the point(5.-9), find the exact value of sec 0 in simplest radical form,

Answer 1

Let us make a simple diagram of the position of the angle using the points (5, -9).

Now, from the right angled triagle in the diagram, let us find the hypotenues, using the Pythagoras theorem.

From the Pythagoras theorem

`\begin{gathered} h^2=5^2+9^2 \\ h^2=25+81 \\ h=\sqrt[]{106} \end{gathered}`

Now,

`\begin{gathered} \cos \theta=\frac{adjacent}{\text{hypotenuse }}=\frac{5}{\sqrt[]{106}} \\ \cos \theta=\frac{5}{\sqrt[]{106}} \\ \text{Also, in this quadrant, cos}\theta\text{ is positive } \end{gathered}`

Then

`\begin{gathered} \sec \theta=(1)/(\cos \theta) \\ \sec \theta=\frac{\sqrt[]{106}}{5} \end{gathered}`

Hence, the answer is

`\sec \theta=\frac{\sqrt[]{106}}{5}`