Let's imagine that there is a garden at a nursery school in the shape of a , and we want to cover this garden with grass. We have the following information about the measures of the garden.
To find out how many square meters of grass is needed, we will find the of this right triangle. However, we are missing the lengths of the . With the measures of two and the length of the , we can find the missing lengths. To do so we can recall the .
asinA=bsinB=csinC
Let's first use the second and third to find the value of
c.
bsinB=csinC
5sin90=csin20
sin90=csin20⋅5
sin90⋅c=sin20⋅5
c=sin90sin20⋅5
c=1.710100…
c≈1.7
Then we will find the
m∠A to find the side length opposite to this angle. Let's use the .
m∠A+90∘+20∘=180∘⇓m∠A=70∘
Now, we can find the value of
a by using the Law of Sines one more time.
asinA=bsinB
asin70=5sin90
asin70⋅5=sin90
sin70⋅5=sin90⋅a
sin90sin70⋅5=a
a=sin90sin70⋅5
a=4.698463…
a≈4.7
Great! Now we can complete our triangle.
Having the lengths of the legs, we can find the area of this right triangle. Let's do it by using the Area Formula!
Area=21⋅base⋅height
Area≈21⋅(4.7)⋅(1.7)
Area≈3.995
Area≈4
We have found that about
4 square meters grass are needed to cover the garden of the nursery school.