a Determine the trigonometric ratio to use according to the given information and the unknown.
B
b The Law of Sines relates the sine of each angle to the length of the opposite side.
C
c Determine the trigonometric ratio to be used according to the information that is given.
A
a x ≈ 8.2
B
b x ≈ 6.32
C
c x ≈ 56.31
Practice makes perfect
a We are given the length of the hypotenuse and the measure of an acute angle of a right triangle. We want to find the length of the other leg.
Note that the given side is the hypotenuse, and the side we want to find is oposite to the given angle. Therefore, we will use the sine ratio.
sin θ = opposite/hypotenuse
In our triangle, we have that θ =27^(∘) and the length of the hypotenuse is 18. We want to find the length of the leg adjacent to the angle.
b For any △ ABC, let the lengths of the sides opposite angles A, B, and C be a, b, and c, respectively.
The Law of Sines relates the sine of each angle to the length of the opposite side.
sin A/a=sin B/b=sin C/c
Let's use this law to find the value of x. Consider the given triangle.
We know that the length of a side is 7 and that the measure of its opposite angle is 102^(∘). We also know that the measure of the angle that is opposite to the side we want to find is 62 ^(∘). With this information and using the Law of Sines, we can write an equation in terms of x.
sin 102^(∘)/7=sin 62^(∘)/x
Let's solve our equation!
c We are given the length of two legs of a right triangle, and want to find the measure of one of its acute angles.
Note that we are given the opposite and the adjacent sides to the unknown angle. Therefore, to find its measure we will use the tangent ratio.
tan x = Length of leg opposite tox/Length of leg adjacent tox
In our triangle, we have that the length of the opposite and adjacent legs to x are 6 and 4.
