Draw a diagram representing the situation. Would you use the Law of Sines or the Law of Cosines to find the distance between Cities B and C?
About 523.8 miles
Practice makes perfect
We are asked to determine the distance between the Cities B and C. To do it, we will first draw a diagram representing the described situation. Then, we will be able to decide which method to use to find the desired distance.
Drawing the Diagram
We know that an airplane flies 55^(∘) east of north from City A to City B, and that the distance between Cities A and B is 470 miles.
Additionally, another airplane flies 7^(∘) north of east from City A to City C, and the distance between Cities A and C is 890 miles.
We can see that the cities are the vertices of a triangle. Note that we know the lengths of two sides of this triangle. The remaining length is the distance d between Cities B and C that we are looking for.
Now, let's calculate the measure of the included angle of the 470-mile and 890-mile sides. Notice that the north and east directions form a right angle, so its measure is 90^(∘).
90^(∘)- 55^(∘)- 7^(∘)=28^(∘)
The measure of the included angle of the 470-mile and 890-mile sides is 28^(∘).
Finding the Distance
Since we know the lengths of two sides of the triangle and the measure of their included angle, we will use the Law of Cosines to find the distance d between Cities B and C. Let's recall the law.
Law of Cosines
If △ ABC has sides of length a, b, and c, then the following are true. a^2=b^2+c^2-2bccosA b^2=a^2+c^2-2accosB c^2=a^2+b^2-2abcosC
We will apply the Law of Cosines to our triangle, and we will use the first equation.
d^2= 890^2+ 470^2-2( 890)( 470)cos 28^(∘)
Let's solve the obtained equation for d.
