Question

question 13 (1 point) saved
△pqr has a right angle at q. if q = 26 cm and p = 10 cm, which of the following is the correct set of reciprocal trigonometric ratios for ∠r?
a)
csc r = \\( \frac { 13 } { 12 } \\), sec r = \\( \frac { 13 } { 5 } \\),
cot r = \\( \frac { 12 } { 5 } \\)
b)
csc r = \\( \frac { 13 } { 5 } \\), sec r = \\( \frac { 13 } { 12 } \\),
cot r = \\( \frac { 5 } { 12 } \\)
c)
csc r = \\( \frac { 12 } { 13 } \\), sec r = \\( \frac { 5 } { 13 } \\),
cot r = \\( \frac { 12 } { 5 } \\)
d)
csc r = \\( \frac { 13 } { 12 } \\), sec r = \\( \frac { 13 } { 5 } \\),
cot r = \\( \frac { 5 } { 12 } \\)

Explanation:

Step1: Find the length of side \( r \)

Using the Pythagorean theorem \( q^{2}=p^{2}+r^{2} \). Given \( q = 26\) and \( p = 10\), we have \( r=\sqrt{q^{2}-p^{2}}=\sqrt{26^{2}-10^{2}}=\sqrt{(26 + 10)(26-10)}=\sqrt{36\times16}=24\).

Step2: Recall the definitions of reciprocal trigonometric ratios

\(\csc R=\frac{q}{p}\) (since \(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}\) for angle \( R\), hypotenuse \( q = 26\), opposite to \( R\) is \( p = 10\)), \(\sec R=\frac{q}{r}\) (since \(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}\) for angle \( R\), hypotenuse \( q = 26\), adjacent to \( R\) is \( r = 24\)), \(\cot R=\frac{r}{p}\) (since \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\) for angle \( R\), adjacent \( r = 24\), opposite \( p = 10\)).
\(\csc R=\frac{26}{10}=\frac{13}{5}\), \(\sec R=\frac{26}{24}=\frac{13}{12}\), \(\cot R=\frac{24}{10}=\frac{12}{5}\)

Answer:

B. \(\csc R=\frac{13}{5},\sec R=\frac{13}{12},\cot R = \frac{12}{5}\)