SINE AND COSINE LAWS
Problem: Levi and Bonnie live one kilometer apart. The school that they attend makes 60-degree angle with the street connecting their houses when measured from Levi’s house. Levi’s house is 3 km away from the school. Find the angle the street connecting Bonnie’s house and the school makes with the street connecting them.
Solution:
As we can see, a triangle is formed as three locations were connected. The three locations are Levi's house, Bonnie's house and their school. Now, let us assign respective letters to label the parts of the triangle.
Let: a = the side of the triangle from Levi's house to Bonnie's house
b = the side of the triangle from Levi's house to their school
c = the side of the triangle from Bonnie's house to their school
C = the angle formed between side a and b
B = the theta in the picture = the angle formed between side a and c
From the problem, we already have three given data: the length of side a and b, and the angle C. Here are their corresponding values:
a = 1 km
b = 3 km
C = 60 degrees
Using cosine law, we can solve for the measure of side c.
[tex]c^2 = a^2 +b^2 -2ab(cosC)[/tex]
By direct substitution:
[tex]c^2 = 1^2 +3^2 -2(1)(3)(cos60)\\c^2 = 1 + 9 - 6cos60\\c^2 = 10 - 6cos60\\c^2 = 10 - 3 = 7\\c = \sqrt{7} = 2.65[/tex]
So the length of side c is square root of 7 kilometers or equivalent to approximately 2.65 km. Now, we can use sine law to solve for the unknown angle.
[tex]\frac{sinB}{b} = \frac{sinC}{c} \\\frac{sinB}{3} = \frac{sin60}{\sqrt{7}}\\sinB = 3 (\frac{sin60}{\sqrt{7}})\\B = sin^{-1} [3 (\frac{sin60}{\sqrt{7}}]\\B = 79.1066[/tex]
Therefore, the angle between Bonnie's house to Levi's house and Bonnie's house to school is equal to 79.11 degrees.
From the choices, I have tried manipulating the values for sin60 and square root of 7 to get answers closest to one of the choices. There is one instance that the answer I calculated get closest to option B. But we stick to the answer we had above, the closest option is option A.
Here are other practice problems related to sine law: https://brainly.ph/question/15905156
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