Problem 4 - Algebra 2

Question

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. \(a=5, b=12, c=13\)

Step-by-Step Solution

Answer

Use Law of Cosines; angles are \(A \approx 23^\circ\), \(B \approx 67^\circ\), \(C = 90^\circ\).

Step 1: Analyze Triangle Data

Given a triangle with sides \(a = 5\), \(b = 12\), \(c = 13\), determine which law to use. Since all the sides are known and no angles are given, we will start with the Law of Cosines.

Step 2: Apply Law of Cosines to Find Angle C

Use the Law of Cosines formula: \[c^2 = a^2 + b^2 - 2ab \cos(C)\]Substitute the known values: \[13^2 = 5^2 + 12^2 - 2(5)(12)\cos(C)\]Calculate and solve for \(\cos(C)\): \[169 = 25 + 144 - 120 \cos(C) \]\[169 = 169 - 120 \cos(C)\]\[0 = -120 \cos(C)\]\[\cos(C) = 0\]Thus, \(C = 90^\circ\).

Step 3: Use Right Triangle Properties

Since \(C = 90^\circ\), this is a right triangle, and angles \(A\) and \(B\) are complementary. Use the Pythagorean identity and known side lengths to verify configurations.

Step 4: Calculate Angles A and B

Use the sine or cosine definitions for a right triangle to find \(A\) and \(B\): \(A + B = 90^\circ\). Since computed earlier \(\cos(C) = 0\), the triangle fits the Pythagorean theorem values, where \(b = 12\) and \(a = 5\). Use: \[\tan(A) = \frac{a}{b} = \frac{5}{12}\]From this ratio, compute angle \(A\): \(A \approx 23^\circ\).This gives \(B \approx 67^\circ\) as this triangle configuration holds for 90, 23, and 67 degrees.

Key Concepts

Law of Cosines

The Law of Cosines is particularly useful for solving triangles when you know all three sides or two sides and the included angle, making it an excellent choice here. It helps in situations where the Law of Sines cannot be directly applied. To find an unknown angle when all three sides are known, the Law of Cosines can be formulated as:

• For angle C: \[c^2 = a^2 + b^2 - 2ab \cos(C)\]

With this formula, you substitute the known values of the sides to solve for the cosine of the angle. Once you have this cosine value, you can determine the angle by taking the inverse cosine. In our example, using sides \(a = 5\), \(b = 12\), and \(c = 13\), we found \(C\) to be \(90^\circ\).

Law of Sines

The Law of Sines is another important trigonometric tool which is ideal when you're given two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

• The formula is: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

This formula allows you to solve for unknown angles and sides by creating a proportion where opposite sides and angles are involved. Yet, it should be noted that our exercise began without any known angles, which is why the Law of Cosines was preferred initially. After finding one angle using the Law of Cosines, the Law of Sines could potentially assist in solving for another side or angle. Still, recognizing a calculation route where you might need this law is a crucial skill.

Right Triangle Properties

A right triangle has one angle measuring \(90^\circ\), simplifying certain calculations significantly. When you recognize a triangle is right-angled, traditional trigonometric identities, such as sine, cosine, and tangent, become particularly useful.

• For example:
  • \(\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}\)
  • \(\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
  • \(\tan(A) = \frac{\text{opposite}}{\text{adjacent}}\)

Also, the angles other than the right angle in this triangle are complementary, meaning their sum equals \(90^\circ\). This complements the Pythagorean relation, making angle and side calculations straightforward once you have identified the right triangle.

Pythagorean Theorem

The Pythagorean Theorem is perhaps one of the most recognizable theorems in mathematics. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

• Expressed as: \[c^2 = a^2 + b^2\]

In our problem, since we determined the triangle is right-angled using the Law of Cosines, we can confirm the side lengths fulfill this equation:

• \(13^2 = 5^2 + 12^2\)

Hence, the triangle is confirmed to be right, and utilizing the knowledge of these lengths and the Pythagorean identity could simplify checking solutions or even deducing other dimensions within the triangle.