Suppose an isosceles triangle [tex]A B C[/tex] has [tex]A=\frac{\pi}{4}[/tex] and [tex]b=c=3[/tex]. What is the length of [tex]a^2[/tex]?
A. [tex]3^2(2-\sqrt{2})[/tex]
B. [tex]3^2(\sqrt{2}-2)[/tex]
C. [tex]3^2(2+\sqrt{2})[/tex]
D. [tex]3^2 \sqrt{2}[/tex]
Answer (2)
To find a 2 in the isosceles triangle, we use the Law of Cosines. After substituting the given values and simplifying, we find that a 2 = 3 2 ( 2 − 2 ) . Thus, the correct answer is A: 3 2 ( 2 − 2 ) .
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Apply the Law of Cosines: a 2 = b 2 + c 2 − 2 b c cos ( A ) .
Substitute the given values: a 2 = 3 2 + 3 2 − 2 ( 3 ) ( 3 ) cos ( 4 π ) .
Simplify the expression using cos ( 4 π ) = 2 2 : a 2 = 18 − 18 ( 2 2 ) = 18 − 9 2 .
Factor out 9 to obtain the final answer: a 2 = 3 2 ( 2 − 2 ) .
Explanation
Problem Analysis We are given an isosceles triangle A BC with angle A = 4 π and sides b = c = 3 . We need to find the length of a 2 .
Applying the Law of Cosines We can use the Law of Cosines to relate the sides and angles of the triangle. The Law of Cosines states: a 2 = b 2 + c 2 − 2 b c cos ( A ) In our case, A = 4 π , b = 3 , and c = 3 .
Substitution Substitute the given values into the Law of Cosines: a 2 = 3 2 + 3 2 − 2 ( 3 ) ( 3 ) cos ( 4 π ) a 2 = 9 + 9 − 18 cos ( 4 π )
Using the value of cosine We know that cos ( 4 π ) = 2 2 , so we can substitute this value into the equation: a 2 = 18 − 18 ⋅ 2 2 a 2 = 18 − 9 2
Simplification We can factor out 9 from the expression: a 2 = 9 ( 2 − 2 ) a 2 = 3 2 ( 2 − 2 )
Final Answer Therefore, the length of a 2 is 3 2 ( 2 − 2 ) .
Examples
The Law of Cosines is a fundamental concept in trigonometry that relates the sides and angles of any triangle. It's particularly useful in fields like surveying and navigation. For example, surveyors use the Law of Cosines to calculate distances and angles in irregular terrains, while navigators use it to determine the position and course of ships or aircraft. Imagine you're designing a triangular garden bed where two sides are each 3 meters long, and the angle between them is 45 degrees. Using the Law of Cosines, you can calculate the length of the third side to determine how much edging you'll need: a 2 = 3 2 + 3 2 − 2 ⋅ 3 ⋅ 3 ⋅ cos ( 4 5 ∘ ) = 9 ( 2 − 2 ) ≈ 5.27 square meters, so a ≈ 2.3 meters.
