c The Law of Cosines relates the cosine of each angle in a triangle to its side lengths.
A
a x≈ 10.7cm
Tool: tangent
B
b x ≈ 7.9mi
Tool: Law of Sines
C
c x ≈ 15.3'
Tool: Law of Cosines
Practice makes perfect
a We are given the length of one leg and the measure of an acute angle of a right triangle. We want to find the length of the other leg.
Notice that the given side is opposite the given angle and that the side we want to find is adjacent to the given angle. Because of this, let's use the tangent ratio to find the missing value x.
tan θ = opposite/adjacent
In our triangle, we have that θ = 40^(∘) and the length of the opposite leg is 9. We want to find the length of the leg adjacent to the angle. Let's do it!
The unknown leg is approximately x ≈ 10.7 centimeters long.
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 its opposite side.
sin A/a=sin B/b=sin C/c
We can use this law to find the missing side length of our triangle. To do so, we will start by drawing a diagram to illustrate the situation. Let's also use y^(∘) to represent the missing angle in the triangle.
We know the measure of the angle opposite the missing side, but we do not know any angle measure and opposite side length pairs in the triangle. However, we are given two out of three angles of the triangle. Let's use the Triangle Angle Sum Theorem to find the angle opposite the 8 -mile side.
63^(∘) + y^(∘) + 52^(∘) = 180^(∘)
y^(∘)=65^(∘)
Now we can add the missing angle measure to our diagram.
Great! Now we know two pairs of angles and their corresponding opposite side lengths. We can use this information to find x, which is the side opposite the 63^(∘) angle. Let's write an equation to relate these measures using the Law of Sines.
sin 63^(∘)/x = sin 65^(∘)/8
Now let's solve our equation!
Since of the lengths were given in miles, our answer is x ≈ 7.9 miles.
c For any △ ABC, the Law of Cosines relates the cosine of each angle to the side lengths of the triangle.
We can use this law to find the values of x. Consider the given triangle.
We know the length of two sides, 12' and 15', and that the measure of their included angle is 68^(∘). We want to use this information to find the length of the third side x'. We can use the Law of Cosines to write an equation in terms of x.
Notice that we only kept the principal root when solving the equation, as x is the length of a side and a length must be nonnegative. Finally, all lengths were given in feet, so our answer is x ≈ 15.3'.
