Problem 15 - Algebra 2 and Trigonometry

Question

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 30,35,45 $$

Step-by-Step Solution

Answer

The angles are approximately 87°, 43°, and 50°.

Step 1: Understand the Problem

We have a triangle with sides measuring 30, 35, and 45. We need to find the measure of each angle of the triangle to the nearest degree.

Step 2: Recall the Law of Cosines

The Law of Cosines formula is given by \( c^2 = a^2 + b^2 - 2ab\cos(C) \) for a triangle with sides \( a \), \( b \), and \( c \), and angle \( C \) between \( a \) and \( b \). We can use this formula to find the angles of the triangle.

Step 3: Calculate the Largest Angle First

Since the largest angle is opposite the longest side, use side 45 to find the largest angle. Substitute \( a = 30 \), \( b = 35 \), and \( c = 45 \) into the Law of Cosines: \[ 45^2 = 30^2 + 35^2 - 2(30)(35)\cos(C) \]Solving this, we find \( \cos(C) \).

Step 4: Solve for \( C \) using a Calculator

Calculate \[ 2025 = 900 + 1225 - 2100\cos(C) \]\[ 2025 = 2125 - 2100\cos(C) \]\[ 2100\cos(C) = 100 \]\[ \cos(C) = \frac{100}{2100} \]\[ \cos(C) = \frac{1}{21} \]Use a calculator to find the angle \( C \): \( C \approx \cos^{-1}(\frac{1}{21}) \approx 87^\circ \).

Step 5: Find the Second Angle Using the Law of Cosines

To find angle \( A \), use \[ 30^2 = 35^2 + 45^2 - 2(35)(45)\cos(A) \]Substitute values and solve for \( \cos(A) \).

Step 6: Calculate Angle \( A \) using a Calculator

Calculate \[ 900 = 1225 + 2025 - 3150\cos(A) \]\[ 900 = 3250 - 3150\cos(A) \]\[ 3150\cos(A) = 2350 \]\[ \cos(A) = \frac{2350}{3150} \]\( A \approx \cos^{-1}(\frac{2350}{3150}) \approx 43^\circ \).

Step 7: Find the Third Angle

Since the sum of angles in a triangle is \(180^\circ\), find the third angle \( B \) as follows: \[ B = 180 - C - A \]\[ B = 180 - 87 - 43 \approx 50^\circ \].

Key Concepts

Triangle Angles

Understanding triangle angles is an important part of geometry. A triangle is a three-sided polygon, and the angles inside it always add up to 180 degrees. When we know the sides of a triangle, finding the angles can seem tricky, but it's a manageable task with the right tools and formulas. For any triangle:

• The largest angle is found opposite the longest side.
• The smallest angle is found opposite the shortest side.
• The angles can be calculated using trigonometric laws like the Law of Cosines or Sines.

These properties help us systematically determine each angle. In our example, the sides measured 30, 35, and 45. Therefore, the first angle to find is the one opposite side 45, which will be the largest. Understanding these basics is crucial when diving into angle calculations.

Trigonometry

Trigonometry plays a vital role in understanding triangles. It's the branch of mathematics that deals with the relationships between the angles and sides of triangles. The Law of Cosines, in particular, is a powerful tool:Given the sides of a triangle, it allows us to find an unknown angle. Its formula is \[ c^2 = a^2 + b^2 - 2ab\cos(C) \]where \( c \) is the side opposite angle \( C \).In the exercise, trigonometry helps us convert side lengths into angles. Why choose the Law of Cosines? Because we know all three sides but none of the angles. This law supports calculations by directly using the sides to define each angle clearly. Solving such exercises enhances our spatial understanding and practical math skills.

Angle Calculation

Calculating angles in a triangle, particularly when given all sides, involves several steps. Here, we utilize the Law of Cosines for precision. Let's break it down:

• First, identify the longest side, and apply the formula to find the largest angle. For instance, if we have sides 30, 35, and 45, start with 45 to find angle \( C \).
• With one angle known, use the formula again or the Law of Sines to deduce another angle. Essentially, repeat the process for any other angles as necessary.
• Finally, calculate the remaining angle using the fact that all angles in a triangle equal 180 degrees.

This method allows us to systematically solve for angles with known side lengths. Practicing this approach improves not only calculation skills but also logical thinking in mathematics.