Question

1. use the diagram below to find each ratio.

(right triangle jlk with right angle at l, kl=21, jk=29, and we need to find sin j, cos j, tan k, sin k)

Explanation:

Step1: Find the length of JL

In right triangle \( JKL \) with \( \angle L = 90^\circ \), \( JK = 29 \), \( KL = 21 \). By Pythagorean theorem \( JL=\sqrt{JK^{2}-KL^{2}}=\sqrt{29^{2}-21^{2}}=\sqrt{(29 + 21)(29 - 21)}=\sqrt{50\times8}=\sqrt{400}=20 \).

Step2: Calculate \( \sin J \)

\( \sin J=\frac{\text{opposite to }J}{\text{hypotenuse}}=\frac{KL}{JK}=\frac{21}{29} \).

Step3: Calculate \( \cos J \)

\( \cos J=\frac{\text{adjacent to }J}{\text{hypotenuse}}=\frac{JL}{JK}=\frac{20}{29} \).

Step4: Calculate \( \tan K \)

\( \tan K=\frac{\text{opposite to }K}{\text{adjacent to }K}=\frac{JL}{KL}=\frac{20}{21} \).

Step5: Calculate \( \sin K \)

\( \sin K=\frac{\text{opposite to }K}{\text{hypotenuse}}=\frac{JL}{JK}=\frac{20}{29} \).

Answer:

\( \sin J=\frac{21}{29} \), \( \cos J=\frac{20}{29} \), \( \tan K=\frac{20}{21} \), \( \sin K=\frac{20}{29} \)