Answer
[tex]\begin{gathered} \sec \theta=\frac{\sqrt[]{53}}{7} \\ \csc \theta=\frac{\sqrt[]{53}}{2} \end{gathered}[/tex]
Explanation
The trigonometric function tangent is defined as follows:
[tex]\tan \theta=\frac{\text{opposite}}{\text{adjacent}}[/tex]
Where
"opposite" refers to the side across the angle θ in the right triangle
"adjacent" refers to the side next to the angle θ in the right triangle
The secant is the reciprocal function of the cosine:
[tex]\sec \theta=\frac{1}{\cos \theta}=\frac{\text{hypotenuse}}{\text{adjacent}}[/tex]
The cosecant is the reciprocal function of the sine:
[tex]\csc \theta=\frac{1}{\sin \theta}=\frac{\text{hypotenuse}}{\text{opposite}}[/tex]
To calculate the cosecant and secant, the first step is to determine the length of the hypothenuse of the triangle.
Considering the given tangent:
[tex]\tan \theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{2}{7}[/tex]
We know that the legs of the triangle have a measure 2units and 7units, using the Pythagorean theorem, we can calculate the length of the hypothenuse:
[tex]a^2+b^2=c^2[/tex][tex]\begin{gathered} 2^2+7^2=c^2 \\ 4+49=c^2 \\ 53=c^2 \\ \sqrt[]{53}=\sqrt[]{c^2} \\ \sqrt[]{53}=c \end{gathered}[/tex]
So,
Opposite=2
Adjacent=7
Hypotenuse =√53
Secant:
[tex]\begin{gathered} \text{sec}\theta=\frac{hypotenuse}{adjacent} \\ \sec \theta=\frac{\sqrt[]{53}}{7} \end{gathered}[/tex]
Cosecant:
[tex]\begin{gathered} \csc \theta=\frac{\text{hypotenuse}}{\text{opposite}} \\ \csc \theta=\frac{\sqrt[]{53}}{2} \end{gathered}[/tex]
