Problem 17 - Precalculus with Limits

Question

In Exercises 5-20, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. \(B = 125^{\circ}40'\), \(a = 37\), \(c = 37\)

Step-by-Step Solution

Answer

The length of side \(b\) and the other angles, \(A\) and \(C\), in the triangle can be calculated using the Law of Cosines and the Law of Sines.

Step 1: Conversion of angle

First, Convert the given angle \(B = 125^{\circ}40'\) to decimal format.

Step 2: Find the length of side \(b\)

Next, use the Law of Cosines formula \(b^2 = a^2 + c^2 - 2ac \cdot \cos(B)\) to find the length of side \(b\).

Step 3: Calculate angle \(A\)

After obtaining the length of side \(b\), use the Law of Cosines again but now with sides \(a\) and \(b\) to calculate angle \(A\). Use the formula \( \cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\) and then use inverse cosine to find angle \(A\).

Step 4: Calculate angle \(C\)

Finally, the last step is to calculate angle \(C\). Since we know that the sum of the angles in a triangle is 180 degrees, we can subtract the sum of angles \(A\) and \(B\) from 180 degrees to find angle \(C\): \(C=180^{\circ} - A - B\).

Key Concepts

Solving Triangles

Triangles are geometric shapes with three sides and three angles. To solve a triangle means finding all its sides and angles. In this context, we discuss using the Law of Cosines, which is helpful when we know:

• Two sides and the included angle
• Three sides

Given that you know two sides of a triangle and the angle between them, you can find the third side using the Law of Cosines. The formula looks like this:\[ b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \] Once you have the third side, you can find the remaining angles using the same principle. This method is a powerful tool in trigonometry to solve any triangle when you don't have a right-angle triangle.

Angle Conversion

Angles can be measured in degrees, minutes, and seconds, but sometimes it's easier to work with angles as decimal numbers. To convert an angle given in minutes and seconds, you transform it into decimal format, which simplifies calculations.For an angle like \(B = 125^{\circ} 40'\), perform the conversion as follows:1 degree is equal to 60 minutes. So, \(40'\) is equal to \(\frac{40}{60} = 0.67\) degrees.Thus, the angle in decimal form is \(125 + 0.67 = 125.67^{\circ}\).This converted value of the angle can then be used in trigonometric calculations more seamlessly.

Triangle Angle Sum

A fundamental property of any triangle is that the sum of its three internal angles always adds up to 180 degrees. When you calculate or measure two angles, you can easily find the third one using:\[ C = 180^{\circ} - A - B \] In problems involving the Law of Cosines, once angles \(A\) and \(B\) are known, finding angle \(C\) becomes straightforward by subtracting the sum of \(A\) and \(B\) from 180 degrees.This principle is simple yet crucial in ensuring that the triangle properties are fulfilled.

Inverse Cosine

The inverse cosine (also written as \(\cos^{-1}\)) is a trigonometric function used to find an angle when you know the cosine value. When solving triangles, if you have calculated the cosine of an angle, you use the inverse cosine to get the angle itself.For instance, once you've found:\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]You can determine angle \(A\) as:\[ A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right) \]This step is essential because it translates the cosine value back into degrees or radians, letting you solve for the specific angle measurement.