Find the measure of [tex]$\angle Q$[/tex], the smallest angle in a triangle whose sides have lengths 4, 5, and 6. Round the measure to the nearest whole degree.
Answer (1)
Identify the smallest angle as opposite the shortest side.
Apply the Law of Cosines: 4 2 = 5 2 + 6 2 − 2 ( 5 ) ( 6 ) cos ( Q ) .
Solve for cos ( Q ) : cos ( Q ) = 60 45 = 0.75 .
Find the angle: Q = arccos ( 0.75 ) ≈ 4 1 ∘ .
Explanation
Identify the smallest angle and the Law of Cosines We are given a triangle with sides of length 4, 5, and 6. We want to find the measure of the smallest angle, which we'll call ∠ Q . The smallest angle is opposite the shortest side, which has length 4. We will use the Law of Cosines to find the angle.
Apply the Law of Cosines The Law of Cosines states that for any triangle with sides a , b , and c , and angle A opposite side a , we have: a 2 = b 2 + c 2 − 2 b c cos ( A ) In our case, a = 4 , b = 5 , c = 6 , and A = Q . So we have: 4 2 = 5 2 + 6 2 − 2 ( 5 ) ( 6 ) cos ( Q )
Solve for cos(Q) Now we solve for cos ( Q ) :
16 = 25 + 36 − 60 cos ( Q ) 16 = 61 − 60 cos ( Q ) 60 cos ( Q ) = 61 − 16 60 cos ( Q ) = 45 cos ( Q ) = 60 45 = 4 3 = 0.75
Find angle Q Now we find the angle Q by taking the inverse cosine (arccos) of 0.75: Q = arccos ( 0.75 ) Using a calculator, we find that: Q ≈ 41.409 6 ∘ Rounding to the nearest whole degree, we get: Q ≈ 4 1 ∘
Final Answer Therefore, the measure of the smallest angle in the triangle is approximately 4 1 ∘ .
Examples
The Law of Cosines is useful in surveying and navigation. For example, if you know the distances between three landmarks, you can use the Law of Cosines to determine the angles formed at each landmark. This allows you to create accurate maps or determine your position relative to the landmarks. Imagine you are sailing and know the distances to three islands. By measuring the angles between the islands, you can pinpoint your location on the map using the Law of Cosines.
