To find the length of the missing side of a right triangle we can use the following trigonometric ratios.
sin θ = opposite side/hypotenuse
cos θ = adjacent side/hypotenuse
tan θ = opposite side/adjacent side
cosec θ = hypotenuse/opposite side
sec θ = hypotenuse/adjacent side
cot θ = adjacent side/opposite side
Example 1 :
Find the measure of each side indicated. Round to the nearest tenth.
Solution :
Hypotenuse side = AB
Opposite side = BC = 4
Adjacent side = AC = x
Here θ is 41°
The sides opposite and adjacent are involving in the trigonometric ratio tan θ
tan θ = Opposite side/Adjacent side = BC/AC
tan 41° = 4/x
0.86 = 4/x
x = 4/0.86
= 4.65
So, the measure of missing side is 4.6.
Example 2 :
Find the measure of each side indicated. Round to the nearest tenth.
Solution :
Hypotenuse side = AB = x
Opposite side = AC = 10.8
Adjacent side = BC
Here θ is 57°
The sides opposite and hypotenuse are involving in the trigonometric ratio sin θ
sin θ = Opposite side/Hypotenuse side = AC/AB
sin 57° = 10.8/x
0.83 = 10.8/x
x = 10.8/0.83
= 13.01
So, the measure of missing side is 13.
Example 3 :
Find the measure of each side indicated. Round to the nearest tenth.
Solution :
Hypotenuse side = AB = 10.3
Opposite side = AC = x
Adjacent side = BC
Here θ is 37°
The sides opposite and hypotenuse are involving in the trigonometric ratio sin θ
sin θ = Opposite side/Hypotenuse side = AC/AB
sin 37° = x/10.3
0.60 = x/10.3
x = 10.3(0.60)
= 6.18
So, the measure of missing side is 6.1.
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