SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Find the value of KM
Using Pythagoras' Theorem,
[tex]\begin{gathered} hypotenuse^2=opposite^2+adjacent^2 \\ hypotenuse=34,adjacent=16,opposite=? \end{gathered}[/tex]
By substitution,
[tex]\begin{gathered} 34^2=opposite^2+16^2 \\ opposite^2=34^2-16^2 \\ opposite^2=900 \\ opposite=\sqrt{900}=30 \end{gathered}[/tex]
KM = 30
STEP 2: Find the required ratio
To get sin L:
[tex]\begin{gathered} \sin\theta=\frac{opp}{hyp} \\ By\text{ substitution,} \\ \sin L=\frac{30}{34}=\frac{15}{17} \end{gathered}[/tex]
sin L = 15/17
To get cos L
[tex]\begin{gathered} \cos\theta=\frac{adjacent}{hypotenuse} \\ \cos L=\frac{16}{34}=\frac{8}{17} \end{gathered}[/tex]
cos L = 8/17
To get tan L
[tex]\begin{gathered} \tan\theta=\frac{opposite}{adjacent} \\ By\text{ substitution,} \\ \tan L=\frac{30}{16}=\frac{15}{8} \end{gathered}[/tex]
tan L = 15/8
STEP 3: Find the ratios for angle M
To get sin M
[tex]\begin{gathered} \sin\theta=\frac{opp}{hyp} \\ By\text{ substitution,} \\ \sin M=\frac{16}{34}=\frac{8}{17} \end{gathered}[/tex]
sin M = 8/17
To get cos M
[tex]\begin{gathered} \cos\theta=\frac{adj}{hyp} \\ By\text{ substitution,} \\ \cos M=\frac{30}{34}=\frac{15}{17} \end{gathered}[/tex]
cos M = 15/17
tan L has been done in Step 2
tan L = 15/8
