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Chapter 5: Problem 19

Use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$C=43^{\circ}, \quad a=\frac{4}{9}, \quad b=\frac{7}{9}$$

Short Answer

Expert verified
After performing the calculations, the dimensions of side c and the measure of the angles A and B rounded to two decimal places will be provided.

Step by step solution

01

Compute the length of side c

The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and an angle C opposite side c, we have: c虏 = a虏 + b虏 - 2ab cos(C). Thus, we can find side c using the given angle C and sides a and b:\n\[ c = sqrt{(\frac{4}{9})^2 + (\frac{7}{9})^2 - 2 * \frac{4}{9} * \frac{7}{9} * cos(43)} \]
02

Calculate angles A and B

To find the missing angles A and B, the Law of Cosines can still be used but in a different form. For example to find angle A: cos(A) = (b虏 + c虏 - a虏) / (2bc). Similarly, for angle B: cos(B) = (a虏 + c虏 - b虏) / (2ac). We also have to make sure that A, B, and C add up to 180掳.
03

Round and write the final results.

After we compute these values they will be rounded to the nearest two decimal places, as requested, to give the dimensions of the triangle's remaining side c and the measures of its missing angles, A and B.

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