Problem 86 - Precalculus

Question

Let \(A, B,\) and \(C\) be the lengths of the three sides with \(X, Y,\) and \(Z\) as the corresponding angle measures in a triangle. Write a program using a TI calculator to solve each triangle with the given measures. $$A=100, B=170, \text { and } C=250$$

Step-by-Step Solution

Answer

Calculate angles using the Law of Cosines and sum of angles property in a triangle.

Step 1: Identify Known Values

We are given the side lengths of the triangle: \(A=100\), \(B=170\), and \(C=250\). These are opposite the angles \(X\), \(Y\), and \(Z\), respectively.

Step 2: Calculate Angle X Using Law of Cosines

We can use the Law of Cosines to find angle \(X\):\[\cos X = \frac{B^2 + C^2 - A^2}{2BC}\]Substitute the given values:\[\cos X = \frac{170^2 + 250^2 - 100^2}{2 \times 170 \times 250}\]Calculate the cosine value and then use the inverse cosine function to find \(X\).

Step 3: Calculate Angle Y

Use the Law of Cosines to find angle \(Y\):\[\cos Y = \frac{A^2 + C^2 - B^2}{2AC}\]Substitute the given values:\[\cos Y = \frac{100^2 + 250^2 - 170^2}{2 \times 100 \times 250}\]Calculate the cosine value and then use the inverse cosine function to find \(Y\).

Step 4: Calculate Angle Z

Since the sum of angles in a triangle is 180 degrees, use:\[Z = 180 - X - Y\]Substitute the values of \(X\) and \(Y\) to find \(Z\).

Step 5: Translate Steps into TI Calculator Program

Create a program on the TI calculator:1. Store the side lengths \(A\), \(B\), \(C\).2. Calculate \(X\) using the formula in Step 2 and the inverse cosine function.3. Calculate \(Y\) using the formula in Step 3.4. Calculate \(Z\) using the formula in Step 4.5. Output the angles \(X\), \(Y\), and \(Z\).

Key Concepts

Law of Cosines

The Law of Cosines is an essential theorem in triangle trigonometry. It helps to find the angles of a triangle when the lengths of all three sides are known. This law is a generalization of the Pythagorean theorem and is used mainly in non-right triangles.
To use the Law of Cosines, the formula for finding an angle can be expressed as follows:

• For angle \(X\) opposite side \(A\): \[\cos X = \frac{B^2 + C^2 - A^2}{2BC}\]
• For angle \(Y\) opposite side \(B\): \[\cos Y = \frac{A^2 + C^2 - B^2}{2AC}\]
• For angle \(Z\) opposite side \(C\): \[\cos Z = \frac{A^2 + B^2 - C^2}{2AB}\]

After calculating the cosine value of an angle using these formulas, the inverse cosine function (often noted as \(\cos^{-1}\) or "arccos") is used to determine the actual angle measure.
It's handy in a variety of practical problems, especially when solving triangles that do not fit a simple right-angled scenario.

Angle Calculation

Once the Law of Cosines provides a cosine value for an angle, calculating the exact angle measure is straightforward. Use the inverse cosine function, which is available on most scientific calculators, to convert the cosine value into degrees. Here’s how it works in practice:

• Determine the cosine with the formula.
• Use the inverse cosine function to find the angle.

For instance, if \(\cos X = 0.5\), use a calculator or TI calculator to compute \(X = \cos^{-1}(0.5)\), which provides an angle in degrees.
This process is repeated for each angle until all angles of the triangle are known. It's a critical skill in trigonometry to transition from decimal or fractional form into practical angle measures that can be applied or visualized in geometry.

TI Calculator Programming

Programming your TI calculator to solve triangles can be a timesaver. It automates repeated calculations and ensures accuracy. Here’s a basic outline of how to create such a program:

• Input Step: First, input or assign the known side lengths \(A\), \(B\), and \(C\) into variables.
• Calculate Angles: Use formulas. Implement the Law of Cosines with your calculator’s programming functions to find \(\cos X\) and then calculate \(X\) using the inverse cosine.
• Repeat for \(Y\) with its respective formula as outlined, and then for \(Z\) by subtracting \(X\) and \(Y\) from 180.
• Output Results: Program the calculator to display the values of angles \(X\), \(Y\), and \(Z\) after calculations.

By structuring this into a calculator script, students can efficiently solve complex triangles without manually processing each calculation, thus minimizing errors and saving time.

Triangle Angle Sum

A fundamental property of triangles, whether in Euclidean geometry or trigonometry, is that the sum of the interior angles equals 180 degrees. This principle is crucial, especially when using methods like the Law of Cosines to determine two angles, and then the third.
Here's why it matters:

• Once two angles are determined, the third is found by subtracting the sum of these angles from 180 degrees. This checks the accuracy of prior calculations.
• It provides an extra verification step to ensure that the calculated angles logically form a triangle.

For example, if angles \(X\) and \(Y\) are calculated as 60 and 70 degrees respectively, then angle \(Z\) can be determined by: \[Z = 180 - 60 - 70\]Thus, \(Z\) would be 50 degrees. This verification ensures the solution adheres to the fundamental rule of triangle geometry.