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Chapter 5: Problem 14

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$a=55, \quad b=25, \quad c=72$$

Short Answer

Expert verified
To solve the triangle, apply the Law osines to find the measures of the angles. Angle A is calculated using the first formula, angle B using the second formula and finally angle C is found by subtracting the sum of A and B from \(180^\circ\). All the obtained angles will be rounded off to two decimal places.

Step by step solution

01

Calculating the first angle

The first angle \(A\) can be determined using the Law of Cosines. The formula for the Law of Cosines is \(a^2 = b^2 + c^2 - 2bc \cos A\). Here, values a=55, b=25, and c=72. By substituting these values into the equation, we receive: \(55^2 = 25^2 + 72^2 - 2*25*72 \cos A\). Solving this for \(\cos A\) and then taking inverse cosine will provide angle A.
02

Calculating the second angle

The second angle \(B\) can be computed using the Law of Cosines as well. Now the formula to use is \(b^2 = a^2 + c^2 - 2ac \cos B\). Insert the known quantities, a=55, b=25, and c=72, into this equation to find: \(25^2 = 55^2 + 72^2 - 2*55*72 \cos B\). Solving this for \(\cos B\) and then taking inverse cosine will provide angle B.
03

Finding the third angle

The third angle can be determined by the knowledge that the sum of all angles in a triangle equals \(180^\circ\). So, subtract the sum of the two computed angles from \(180^\circ\) to obtain the third angle. \(C = 180^\circ - A - B\).

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