Option A:
[tex]$\cos M=\frac{19}{20}[/tex]
Solution:
The image of the triangle is attached below.
Given data:
[tex]$\sin L=\frac{19}{20}[/tex]
Using trigonometric ratio formulas,
[tex]$\sin L=\frac{\text { Opposite side of } L}{\text { Hypotenuse }}[/tex]
[tex]$\sin L=\frac{MN}{LM}[/tex]
So, MN = 19 and LM = 20
Using Pythagoras theorem,
In right triangle, square of the hypotenuse is equal to the sum of the squares of the other two sides.
[tex]LM^2=LN^2+MN^2[/tex]
[tex]20^2=LN^2+19^2[/tex]
[tex]400=LN^2+361[/tex]
Subtract 361 from both sides,
39 = LN²
Taking square root on both sides,
[tex]LN=\sqrt{39}[/tex]
[tex]$\cos \theta=\frac{\text { Adjacent side of } \theta}{\text { Hypotenuse }}[/tex]
[tex]$\cos M=\frac{\text { Adjacent side of } M}{\text { Hypotenuse }}[/tex]
[tex]$\cos M=\frac{MN}{LM}[/tex]
[tex]$\cos M=\frac{19}{20}[/tex]
Hence option A is the correct answer.
