Question

Given sec A = √65/ 7 and that angle A is in Quadrant I, find the exact value of csc A in simplest radical form using a rational denominator.

Answer 1

First, notice that we can write sec(A) like this:

$\sec (A)=\frac{1}{\cos (A)}$

then, given the information on the problem, we have the following:

$\begin{gathered} \sec (A)=\frac{\sqrt[]{65}}{7} \\ \Rightarrow\frac{1}{\cos (A)}=\frac{\sqrt[]{65}}{7} \\ \Rightarrow\cos (A)=\frac{7}{\sqrt[]{65}} \end{gathered}$

We also know that the cosine of an angle is defined as the opposite side divided by the hypotenuse of a right triangle. We can see this in the following picture:

then, we can find the missing side using the pythagorean theorem:

$\begin{gathered} (\sqrt[]{65})^2=x^2+(7)^2 \\ \Rightarrow x^2=65-(7)^2=65-49=16 \\ \Rightarrow x=\sqrt[]{16}=4 \\ x=4 \end{gathered}$

now that we found the measure of the opposite side of angle A,we can calculate the sine of A to get:

$\sin (A)=\frac{\text{opposite side}}{hypotenuse}=\frac{4}{\sqrt[]{65}}$

then, we have the following property:

$\csc (A)=\frac{1}{\sin (A)}$

thus, using the value that we found for sin(A), we have:

$\csc (A)=\frac{1}{\sin (A)}=\frac{1}{\frac{4}{\sqrt[]{65}}}=\frac{\sqrt[]{65}}{4}$

therefore, csc(A)=sqrt(65)/4