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Chapter 5: Problem 22

Round answers according to the rounding conventions on page 364. Find the third side of the triangle. $$a=25.9, c=33.4, B=84.0^{\circ}$$

Short Answer

Expert verified
The length of the third side \(b\) can be calculated using the Law of Cosines. Substitute the given values into the formula, then compute and take the square root of \(b^2\), and finally apply the rounding conventions.

Step by step solution

01

Understanding the Law of Cosines

The Law of Cosines, can be formulated as: \(b^2 = a^2 + c^2 - 2ac \cdot cos(B)\) where a and c are the sides surrounding angle B.
02

Substituting the given values

Take the given side lengths and angle, and substitute the values into the Law of Cosines equation. Thus, our equation becomes \(b^2 = (25.9)^2 + (33.4)^2 - 2*(25.9)*(33.4) \cdot cos(84)\)
03

Calculating the missing side length

After performing the required operations and applying the cosine part using a calculator set in degree, the left side of the equation becomes a single value, \(b^2\). To solve for \(b\), we need to take the square root of \(b^2\). Be sure to consider the positive root because lengths cannot have a negative value.
04

Rounding according to conventions

The final step involves rounding the number obtained from the square root operation according to the conventions mentioned on page 364. Remember, you should follow the specific rounding rules defined there.

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