Using 3 different methods, we can find that YZ = 25m
For a given triangle rectangle with a known angle θ and a hypotenuse H, we can write the trigonometric relations
sin(θ) = (opposite cathetus)/H
cos(θ) = (adjacent cathetus)/H
tan(θ) = (opposite cathetus)/(adjacent cathetus)
Now, in the given image we can see that:
θ = 30°
H = 50m
XY = adjacent cathetus
YZ = opposite cathetus.
We want to find YZ, so with the known things, we can use the first relation:
[tex]sin(30\°) = YZ/50m[/tex]
[tex]sin(30\°)*50m = YZ = 25m[/tex]
Now let's try another way.
We know that the sum of the internal angles of a triangle is always equal to 180°
In a triangle rectangle, we always have an angle equal to 90°.
Then the missing angle for our triangle can be computed from:
Z + 90° + 30° = 180°
Z = 180° - 90° - 30° = 60°
From this angle, the side YZ is the adjacent cathetus, then we can use the second trigonometric relation:
[tex]cos(60\°) = YZ/50m\\cos(60\°)*50m = YZ = 25m\\[/tex]
Third way.
Whit the same angle we could find the side XY, the opposite cathetus, with:
[tex]sin(60\°) = XY/50m \\sin(60\°)*50m = XY = 43.3m[/tex]
Now that we know one of the catheti, we can use the last trigonometric relation
[tex]tan(60\°) = XY/YZ = 43.3m/YZ \\YZ = 43.3m/tan(60\°) = 25m\\[/tex]
If you want to learn more about triangle rectangles, you can read:
https://brainly.com/question/24330420
