7. In a triangular plot, two sides are 70 meters and 50 meters long and the angle between them is 60 degrees. What is the length of the third side?
8. Solve for 227i expressing your answer in the Cartesian form.
Answer (1)
Use the law of cosines: c 2 = a 2 + b 2 − 2 ab cos ( C ) to find the length of the third side of the triangle.
Substitute the given values: a = 70 , b = 50 , and C = 6 0 ∘ into the formula.
Calculate c 2 = 7 0 2 + 5 0 2 − 2 ( 70 ) ( 50 ) cos ( 6 0 ∘ ) = 3900 .
Find the length of the third side: c = 3900 ≈ 62.45 meters and express 227 i in Cartesian form: 0 + 227 i .
Explanation
State the Law of Cosines We are given a triangle with two sides of lengths 70 meters and 50 meters, and the angle between them is 60 degrees. We need to find the length of the third side. We can use the law of cosines to solve this problem. The law of cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c 2 = a 2 + b 2 − 2 ab cos ( C )
Substitute the given values In our case, we have a = 70 meters, b = 50 meters, and C = 60 degrees. We can substitute these values into the law of cosines formula:
c 2 = 7 0 2 + 5 0 2 − 2 ( 70 ) ( 50 ) cos ( 6 0 ∘ )
Simplify the equation We know that cos ( 6 0 ∘ ) = 0.5 , so we can further simplify the equation:
c 2 = 7 0 2 + 5 0 2 − 2 ( 70 ) ( 50 ) ( 0.5 ) c 2 = 4900 + 2500 − 3500 c 2 = 7400 − 3500 c 2 = 3900
Find the square root Now, we take the square root of both sides to find the length of the third side, c:
c = 3900 c ≈ 62.45 meters
Express in Cartesian form Next, we need to solve for 227 i and express the answer in Cartesian form. The Cartesian form of a complex number is a + bi , where a is the real part and b is the imaginary part. In this case, we have 227 i , which can be written as 0 + 227 i .
State the Cartesian form Therefore, the Cartesian form of 227 i is 0 + 227 i .
Final Answer The length of the third side of the triangle is approximately 62.45 meters, and the Cartesian form of 227 i is 0 + 227 i .
Examples
The law of cosines is a fundamental concept in trigonometry and has many real-world applications. For example, it can be used in surveying to calculate distances and angles in land plots, in navigation to determine the position of a ship or aircraft, and in engineering to design structures and mechanisms. Imagine you're designing a bridge, and you need to calculate the length of a support beam. If you know the lengths of two adjacent beams and the angle between them, you can use the law of cosines to accurately determine the required length of the support beam, ensuring the bridge's stability and safety.
