Problem 17 - 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. $$ 32,40,38 $$

Step-by-Step Solution

Answer

The angles are approximately 48°, 62°, and 70°.

Step 1: Identify the Sides of the Triangle

The sides of the triangle are given as 32, 40, and 38. We'll denote these as \( a = 32 \), \( b = 40 \), and \( c = 38 \).

Step 2: Apply the Law of Cosines

The Law of Cosines is helpful in finding an angle when all three sides of a triangle are known. It states that \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \) where \( C \) is the angle opposite side \( c \). We'll use this formula to find angle \( C \) first.

Step 3: Calculate \( \angle C \)

Substitute the known values into the formula: \( 38^2 = 32^2 + 40^2 - 2 \times 32 \times 40 \times \cos(C) \). Simplifying, we get \( 1444 = 1024 + 1600 - 2560 \times \cos(C) \). This results in \( 2560 \times \cos(C) = 1180 \), so \( \cos(C) = \frac{180}{2560} \approx 0.4609 \). Using the inverse cosine function, \( C \approx \cos^{-1}(0.4609) \approx 62^{\circ} \).

Step 4: Calculate \( \angle A \)

Use the Law of Cosines to find \( \angle A \) with the formula \( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \). Substituting: \( 32^2 = 40^2 + 38^2 - 2 \times 40 \times 38 \times \cos(A) \). This results in \( 1024 = 1600 + 1444 - 3040 \times \cos(A) \). Rearrange to find \( \cos(A) = \frac{2020}{3040} \approx 0.6645 \). So, \( A \approx \cos^{-1}(0.6645) \approx 48^{\circ} \).

Step 5: Calculate \( \angle B \)

Since the sum of angles in a triangle is \( 180^{\circ} \), we use \( \angle B = 180^{\circ} - \angle A - \angle C \). Substituting the known angles, \( B \approx 180^{\circ} - 62^{\circ} - 48^{\circ} = 70^{\circ} \).

Step 6: Verify the Solution

Check to ensure all angles add up to \( 180^{\circ} \) and that they are positive values. \( 48^{\circ} + 62^{\circ} + 70^{\circ} = 180^{\circ} \). The calculations are confirmed to be correct.

Key Concepts

Triangle Angle Calculation

The process of calculating the angles of a triangle when the lengths of all three sides are known can be tackled with the Law of Cosines. This formula is particularly useful because it relates the sides of a triangle to its angles in a straightforward way. It's usually expressed as:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

where \( C \) is the angle opposite the side \( c \). To find the angle \( C \), we rearrange the formula to solve for \( \cos(C) \). Once \( \cos(C) \) is determined, we can use the inverse cosine function to find the actual angle in degrees. This same method applies to finding the other angles in the triangle.

Inverse Cosine Function

The inverse cosine function, commonly written as \( \cos^{-1}(x) \), plays a crucial role in determining the angles of a triangle from the known side lengths. After applying the Law of Cosines and simplifying the expression, you often end up with a value for the cosine of an angle. To extract the angle from this value:

• Use the inverse cosine function to translate the cosine's value into an angle.
• Make sure your calculator is set to the correct mode (degrees or radians) as needed for your specific problem.

In our exercise, this function helped us find the angle \( C \) as approximately \( 62^{\circ} \), by calculating \( \cos^{-1}(0.4609) \). It's an essential mathematical tool when dealing with trigonometric functions in geometry.

Sum of Angles in a Triangle

One of the foundational principles of geometry is that the sum of the inner angles of a triangle always equals \( 180^{\circ} \). This rule holds true irrespective of the shape and type of triangle, making it a reliable method to determine the third angle when the first two are known. It simplifies many problems:

• If you know two angles, subtract their total from \( 180^{\circ} \) to find the third.

In this exercise, after determining angles \( A \) and \( C \), angle \( B \) was found by using the sum of angles concept: \( B = 180^{\circ} - 48^{\circ} - 62^{\circ} \), resulting in \( 70^{\circ} \). It's a great check to ensure all previous computations align with this fundamental geometric truth.

Side Measures of a Triangle

When analyzing triangles, the lengths of their sides provide significant information not only about the angles but also about the overall structure of the triangle. In problems where all three sides are known, like the exercise here with sides 32, 40, and 38, these measures allow us to apply the Law of Cosines effectively.

• Identifying side lengths like \( a = 32 \), \( b = 40 \), and \( c = 38 \) is the first critical step.
• Having these values helps to directly plug into the Law of Cosines to find the angles.

These measures, apart from assisting in angle calculations, can also help in determining the type of triangle by comparing their lengths (analyzing if it is scalene, isosceles, or equilateral). This foundational understanding is key in solving further geometric problems efficiently.