Problem 3 - Precalculus : Building Concepts and Connections

Question

Use the Law of Cosines to find the umknoten side of each of the given triangles. The standard notation for labeling of triangles is used. Round anstoers to four decimal places. $$a=12, c=8, B=56^{\circ}$$

Step-by-Step Solution

Answer

The umknoten side b of the triangle, rounded to four decimal places, is the result from Step 4.

Step 1: Write down the Law of Cosines

The Law of Cosines states that in any triangle with sides of lengths a, b, and c, and opposite angles A, B, and C respectively, the following equation is true: \[c^2 = a^2 + b^2 - 2ab \cos C\] For the given triangle, we rearrange this formula to solve for b: \[b^2 = a^2 + c^2 - 2ac \cos B\]

Step 2: Substitute the known values

Next, substitute the known values into the equation. This gives you: \[b^2 = 12^2 + 8^2 - 2 * 12 * 8 * \cos 56^\circ \]

Step 3: Evaluate cos B and solve

Evaluate the cosine of 56 degrees. Then, compute the remaining arithmetic to solve for b^2. Finally, take the square root of both sides to solve for b.

Step 4: Round the answer to four decimal places

Once the numerical value for b is computed, round the answer to four decimal places as instructed.

Key Concepts

triangle sides

When discussing triangles, we label the sides based on their relation to the angles. Recognizing these sides is essential when applying trigonometry, particularly the Law of Cosines. In any triangle, there are three sides, typically labeled as \(a\), \(b\), and \(c\). These sides are opposite their respective angles, labeled \(A\), \(B\), and \(C\).

For example, in a triangle where \(a = 12\), \(c = 8\), and angle \(B = 56^{\circ}\), each side is opposite its corresponding angle; \(a\) is opposite \(A\), \(b\) is opposite \(B\), and \(c\) is opposite \(C\).

• The side opposite the angle you’re dealing with is crucial in applying the Law of Cosines.
• These side and angle labels help to substitute values correctly in trigonometric equations.

Understanding these relationships helps in correctly applying trigonometric policies such as the Law of Cosines, which connects the sides to their opposite angles.

cosine function

The cosine function is a fundamental element in trigonometry and is intimately connected with right and non-right triangles. This function relates the angle of a triangle to the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

In more complex triangles, like the one described in the exercise, the cosine of an angle provides the key to calculating unknown side lengths. In our scenario with angle \(B = 56^{\circ}\), we use this function to find \(b\) with the modified Law of Cosines formula:
\[b^2 = a^2 + c^2 - 2ac \cos B\]
To do this:

• Calculate \(\cos 56^{\circ}\), a necessary step for substitution into the equation.
• This cosine value will help determine the squared length of the side \(b\).

The result of the cosine function affects how we determine the exact length, underlining the importance of correctly finding \(\cos B\) to solve for \(b\).

trigonometry

Trigonometry is a branch of mathematics focused on angles, sides, and the relationships between them. It offers tools like the Law of Cosines, which is vital for solving problems involving non-right triangles.

The Law of Cosines is particularly useful when you know two sides and an included angle or all three sides. For our case, it's directly applied with the equation:
\[b^2 = a^2 + c^2 - 2ac \cos B\]
Here’s how trigonometry helps:

• Defines Relationships: It establishes exact relationships between angles and sides in triangles.
• Expands Solution Methods: Provides ways to solve for unknown sides when complete direct measurement is not possible.

By substituting known values and using trigonometric functions like cosine, we derive precise values for unknown sides. This approach showcases the practicality of trigonometry in various geometric problems, making it an indispensable mathematical tool.