Answer:
Here we have two triangle rectangles, and we want to know the value of y, the shared cathetus between them.
First, we know that one of them has an angle of 60° and the adjacent cathetus to it is a.
The other has an angle of 30° and the adjacent cathetus is b.
In both cases, the opposite cathetus is y.
Now we can recall the relationship:
Tan(a) = (opposite cathetus)/(adjacent cathetus)
Then we can write:
tan(60°) = y/a
tan(30°) = y/b
And we also know that:
a + b = 15
Then we have 3 equations:
tan(60°) = y/a
tan(30°) = y/b
a + b = 15
To solve this, we first need to isolate one variable in one of the equations, i will isolate a in the third equation:
a = 15 - b
Now we can replace it in the first equation to get:
tan(60°) = y/(15 - b)
i will rewrite this one as:
tan(60°)*(15 - b) = y
then our system is:
tan(60°)*(15 - b) = y
tan(30°) = y/b
now we can isolate b in the second equation to get:
b = y/tan(30°)
and then replace it on the other equation to get:
tan(60°)*(15 - y/tan(30°)) = y
Now we can remember that:
tan(60°) = √3
tan(30°) = (√3)/3
Replacing these in our equation we get:
√3*(15 - y/((√3)/3)) = y
we can rewrite this as:
√3*(15 - y*3/√3) = y
then:
(√3)*15 - y*3 = y
(√3)*15 = y + y*3
(√3)*15 = 4*y
(√3)*15/4 = y
then we get: