Problem 14 - Algebra 2 and Trigonometry

Question

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle A B C, b=12.4, c=8.70,\) and \(\mathrm{m} \angle A=23\)

Step-by-Step Solution

Answer

Use the formula \( a^2 = b^2 + c^2 - 2bc\cos(A) \) to find \( a \); the third side is approximately \( 5.2 \).

Step 1: Understanding the Problem

We need to find the measure of the third side of triangle \( \triangle ABC \), where \( b = 12.4 \), \( c = 8.70 \), and \( \angle A = 23^\circ \). We will use the Cosine Law because we have two sides and the included angle.

Step 2: Applying the Law of Cosines

The Cosine Law is given by \( a^2 = b^2 + c^2 - 2bc\cos(A) \). We need to substitute \( b = 12.4 \), \( c = 8.70 \), and \( \cos(23^\circ) \) into this formula to find \( a \).

Step 3: Calculating Values

First, calculate \( b^2 = 12.4^2 = 153.76 \) and \( c^2 = 8.70^2 = 75.69 \). Then calculate the cosine component: \( 2bc\cos(A) = 2 \times 12.4 \times 8.70 \times \cos(23^\circ) \). Use a calculator for \( \cos(23^\circ) \).

Step 4: Calculate \( a^2 \)

Substitute the values into the Cosine Law formula: \[ a^2 = 153.76 + 75.69 - 2 \times 12.4 \times 8.7 \times \cos(23^\circ) \].

Step 5: Solving for \( a \)

Calculate \( 2 \times 12.4 \times 8.7 \times \cos(23^\circ) \), then substitute in and simplify to find \( a^2 \). Take the square root of the result to find \( a \), rounding to the nearest tenth.

Step 6: Conclusion

The length of the third side, \( a \), calculated using the above steps is the answer, rounding to one decimal place.

Key Concepts

Understanding Triangle Side Calculation

Calculating the unknown side of a triangle, especially when dealing with non-right triangles, is an essential skill in geometry. In this case, we apply the Law of Cosines, which is particularly helpful when we know two sides and the included angle of a triangle. This is a scenario where the traditional Pythagorean theorem doesn’t work as it only applies to right triangles.
The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula can be written as:

• \( a^2 = b^2 + c^2 - 2bc\cos(A) \)

Here, \( a \) is the side opposite to angle \( A \), and \( b \) and \( c \) are the other two sides of the triangle. Applying this formula requires substituting the known values into it and solving for the unknown side.
Begin by calculating the squares of the sides you already know, as well as the product of these sides with the cosine of the included angle. The final step is taking the square root of your result to find the length of the missing side, ensuring you round to the nearest tenth for precision.

Included Angle in Trigonometry

The included angle in a triangle is the angle formed between two known sides. In trigonometry, understanding the included angle is crucial as it influences both geometric calculations and trigonometric identities.
In our current triangle scenario, the angle given is part of the critical role in using the Law of Cosines because it determines the trigonometric conversion needed for calculating the unknown side. When using the cosine function, the specific angle needs to be plugged into the equation to find the correct component of \( 2bc\cos(A) \).
Using technology, such as a calculator, allows us to determine \( \cos(θ) \) for the included angle, even if it's not a standard angle. This approach ensures that you utilize accurate values and achieve precise calculations for subsequent steps.
The precision of this angle measurement directly impacts the accuracy of your triangle calculations, underscoring the importance of careful angle evaluation.

Basics of Trigonometry in Triangle Calculations

Trigonometry is a branch of mathematics that deals with relationships between side lengths and angles of triangles. It is fundamental when engaging with triangle calculations like the ones involving the Law of Cosines.
In triangle problems, trigonometrics allow us to move beyond simple side measurements. Trigonometric functions like sine, cosine, and tangent provide powerful tools for solving problems involving non-right triangles, which you encounter frequently in real-world problems.
When solving for triangle side calculations using trigonometry:

• Understand which trigonometric rule or law best applies to your situation. For non-right triangles, the Law of Sines or Cosines is used.
• Use identities, like the cosine rule, to relate angles and sides.
• Make use of calculators for trigonometric values, especially when dealing with non-standard angles.

Engaging deeply with trigonometry in scenarios of triangles solidifies your mathematical understanding and offers a gateway to more complex geometric problems. In practical terms, developing a strong intuition for how these trigonometric concepts interlink with sides and angles of triangles will fine-tune your problem-solving skills.