# Question #76773

May 29, 2017

1. $\frac{3131}{3744} \approx 0.8363$
2. $\frac{181}{60} \approx 3.017$

#### Explanation:

You are given that $\cos \left(A\right) = \frac{5}{13}$. That is, for a triangle, the adjacent, "adj", angle is $5$ and the hypotenuse, "hyp", is $13$.

Using the Pythagorean Theorem, you can get the opposite side

${\text{adjacent"^2+"opposite"^2="hypotenuse}}^{2}$

${\left(5\right)}^{2} + {\text{opposite}}^{2} = {\left(13\right)}^{2}$

$25 + {\text{opp}}^{2} = 169$

${\text{opp}}^{2} = 144$

$\text{opp} = 12$

Using the right-triangle rules of trigonometric functions we have:

$\sin \left(A\right) = \left(\text{opp")/("hyp}\right) = \frac{12}{13}$
$\cos \left(A\right) = \left(\text{adj")/("hyp}\right) = \frac{5}{13}$
$\tan \left(A\right) = \left(\text{opp")/("adj}\right) = \frac{12}{5}$
$\csc \left(A\right) = \left(\text{hyp")/("opp}\right) = \frac{13}{12}$
$\sec \left(A\right) = \left(\text{hyp")/("adj}\right) = \frac{13}{5}$
$\cot \left(A\right) = \left(\text{adj")/("opp}\right) = \frac{5}{12}$

Plug these values into the first question:

$\sin \left(A\right) - \cot \frac{A}{2 \tan \left(A\right)} = \frac{12}{13} - \frac{\frac{5}{12}}{2 \times \frac{12}{5}} = \frac{12}{13} - \frac{5}{12} \times \frac{5}{24}$

$= \frac{12}{13} - \frac{25}{288} = \frac{3131}{3744} \approx 0.8363$

Finally, plug our values into the second question:

$\cot \left(A\right) + \frac{1}{\cos \left(A\right)} = \frac{5}{12} + \frac{1}{\frac{5}{13}} = \frac{5}{12} + \frac{13}{5} = \frac{181}{60} \approx 3.017$