6. Evaluate: tanh 4x
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?
Answer (1)
Evaluate t anh ( 4 x ) using the definition of hyperbolic tangent: t anh ( 4 x ) = e 4 x + e − 4 x e 4 x − e − 4 x .
Apply the Law of Cosines to find the third side of the triangle: c 2 = a 2 + b 2 − 2 ab cos ( C ) .
Substitute the given values a = 70 , b = 50 , and C = 6 0 ∘ into the Law of Cosines: c 2 = 7 0 2 + 5 0 2 − 2 ( 70 ) ( 50 ) cos ( 6 0 ∘ ) = 3900 .
Calculate the length of the third side: c = 3900 ≈ 62.45 meters. 62.45 meters
Explanation
Evaluate tanh 4x The first question asks us to evaluate t anh ( 4 x ) . Recall the definition of the hyperbolic tangent function: t anh ( x ) = cos h ( x ) s inh ( x ) = e x + e − x e x − e − x . Therefore, t anh ( 4 x ) = e 4 x + e − 4 x e 4 x − e − 4 x .
Apply the Law of Cosines The second question involves finding the length of the third side of a triangle given two sides and the included angle. We can use the Law of Cosines to solve this. Let the sides be a = 70 meters, b = 50 meters, and the angle between them be C = 60 degrees. The Law of Cosines states: c 2 = a 2 + b 2 − 2 ab cos ( C ) .
Substitute the values Substitute the given values into the Law of Cosines: c 2 = 7 0 2 + 5 0 2 − 2 ( 70 ) ( 50 ) cos ( 6 0 ∘ ) . Since cos ( 6 0 ∘ ) = 2 1 , the equation becomes: c 2 = 4900 + 2500 − 2 ( 70 ) ( 50 ) ( 2 1 ) = 4900 + 2500 − 3500 = 3900 .
Calculate the third side Solve for c : c = 3900 = 10 39 ≈ 62.45 meters.
Examples
The hyperbolic tangent function, t anh ( x ) , is used in various fields such as physics, engineering, and machine learning. For example, it appears in the study of the shapes of hanging cables and the analysis of neural networks. The Law of Cosines is a fundamental concept in trigonometry and is used in surveying, navigation, and engineering to calculate distances and angles in triangles. For instance, if you are a surveyor and need to determine the distance across a lake, you can measure the lengths of two sides of a triangle and the angle between them, and then use the Law of Cosines to calculate the unknown distance.
