Problem 17 - Algebra 2 and Trigonometry

Question

In \(14-19,\) find, to the nearest tenth, the measure of the third side of each triangle. In \(\triangle A B C, a=62.5, b=44.7,\) and \(\mathrm{m} \angle C=133\)

Step-by-Step Solution

Answer

The third side \( c \approx 98.9 \).

Step 1: Understanding the Problem

In triangle \( \triangle ABC \), we are given two sides \( a = 62.5 \) and \( b = 44.7 \), and an angle opposite the third side, \( \angle C = 133^\circ \). We need to find the third side, \( c \), using these values.

Step 2: Apply the Law of Cosines

The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab \cos(C) \). Given \( a = 62.5 \), \( b = 44.7 \), and \( C = 133^\circ \), we plug these into the formula: \[ c^2 = 62.5^2 + 44.7^2 - 2 \times 62.5 \times 44.7 \times \cos(133^\circ). \]

Step 3: Calculate Cosine of Angle C

Using a calculator, \( \cos(133^\circ) \approx -0.6947 \). Substitute this value into the equation \( c^2 = 62.5^2 + 44.7^2 - 2 \times 62.5 \times 44.7 \times (-0.6947) \).

Step 4: Compute the Squares and Product

Calculate the squares of the sides and the product: \( 62.5^2 = 3906.25 \) and \( 44.7^2 = 1995.29 \). Now calculate the product: \( 2 \times 62.5 \times 44.7 \times (-0.6947) = -3878.36 \). The equation becomes \( c^2 = 3906.25 + 1995.29 + 3878.36 \).

Step 5: Solve for c^2 and Calculate c

Combine the terms: \( c^2 = 3906.25 + 1995.29 + 3878.36 = 9780 \). Take the square root to find \( c \): \( c = \sqrt{9780} \approx 98.9 \).

Step 6: Round to the Nearest Tenth

Since \( c \approx 98.9 \) is already rounded to the nearest tenth, we use this value.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is a crucial tool in fields such as physics, engineering, and astronomy. In trigonometry, we use various functions, like sine, cosine, and tangent, to link the angles to the sides of right-angled triangles, and extend these concepts to non-right-angled triangles.

The Law of Cosines, which is used in this exercise, is a fundamental part of trigonometry. This law helps us solve triangles that are not necessarily right-angled. The formula, \( c^2 = a^2 + b^2 - 2ab \cos(C) \), connects the lengths of the sides to one of the triangle's angles. Understanding and applying this formula allows us to find unknown sides or angles, which is the essence of solving triangles in trigonometry.

Solving Triangles

Solving triangles involves finding unknown angles or sides of a triangle when certain other elements (sides or angles) are given. This often requires using the Laws of Sines and Cosines.

In this example, we used the Law of Cosines because we knew two sides, \( a = 62.5 \) and \( b = 44.7 \), and an included angle, \( C = 133^\circ \), helping us find the unknown side, \( c \). Solving such a triangle means applying the formula \( c^2 = a^2 + b^2 - 2ab \cos(C) \) and plugging in the known values to calculate \( c \).

Each triangle problem is unique, and learning to identify which law or formula to apply is vital. As you practice, you'll be able to solve triangles more efficiently by recognizing patterns in the given data.

Angle Measurement

Measuring angles accurately is a vital part of trigonometry. Angle measurement is usually expressed in degrees, though radians are also commonly used in higher mathematics.

Here, we measure angle \( C \) as \( 133^\circ \). This angle is important because it helps us calculate \( \cos(C) \), a step crucial for applying the Law of Cosines.

Most scientific calculators allow you to convert angles and compute values like \( \cos \), \( \sin \), and \( \tan \). In this exercise, finding \( \cos(133^\circ) \approx -0.6947 \) was necessary. Understanding how to compute these functions enables you to work through solving triangles using trigonometry fully. Keep in mind that precision in your calculations can significantly impact your results, so it's essential to ensure your calculator is set to the correct angle unit, either degrees or radians.