Problem 32 - A Graphical Approach to Precalculus with Limits

Question

Solve each triangle. \(C=59.70^{\circ}, a=3.725\) miles, \(b=4.698\) miles

Step-by-Step Solution

Answer

Side \(c\approx 4.388\) miles, angle \(A\approx 47.12^{\circ}\), angle \(B\approx 73.18^{\circ}\).

Step 1: Identify the Triangle Type

In this exercise, we know two sides and the included angle (SAS situation). Therefore, we need to use the Law of Cosines to solve the triangle.

Step 2: Apply the Law of Cosines

Using the Law of Cosines, we can find the third side \(c\). The formula is \(c^2 = a^2 + b^2 - 2ab \cos(C)\). Fill in the known values:\[c^2 = 3.725^2 + 4.698^2 - 2 \cdot 3.725 \cdot 4.698 \cdot \cos(59.70^{\circ})\]

Step 3: Calculate Side c

Calculate each part of the formula: - \(3.725^2 = 13.8656\)- \(4.698^2 = 22.0788\)- \(2 \cdot 3.725 \cdot 4.698 \cdot \cos(59.70^{\circ})\) is approximately 16.6934Therefore, \[c^2 \approx 13.8656 + 22.0788 - 16.6934 = 19.2509\]\[c \approx \sqrt{19.2509} \approx 4.388 \text{ miles}\]

Step 4: Use the Law of Sines

Now that we have all three sides, we can find the other angles using the Law of Sines:\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]Start with angle \(A\) using \(a\) and \(C\):\[\sin(A) = \frac{a \cdot \sin(C)}{c} = \frac{3.725 \cdot \sin(59.70^{\circ})}{4.388}\]

Step 5: Calculate Angle A

Evaluate the sine and solve for \(A\):- \(\sin(59.70^{\circ}) \approx 0.8625\)\[\sin(A) \approx \frac{3.725 \cdot 0.8625}{4.388} \approx 0.7323\]Use the inverse sine function to find \(A\):\[A \approx \sin^{-1}(0.7323) \approx 47.12^{\circ}\]

Step 6: Calculate Angle B

Now use the sum of angles in a triangle to find \(B\):\[B = 180^{\circ} - C - A = 180^{\circ} - 59.70^{\circ} - 47.12^{\circ}\]\[B \approx 73.18^{\circ}\]

Step 7: Summarize the Triangle

The solved triangle has the following measurements:\(c \approx 4.388\) miles, \(A \approx 47.12^{\circ}\), \(B \approx 73.18^{\circ}\), and given \(C = 59.70^{\circ}\), \(a = 3.725\) miles, and \(b = 4.698\) miles.

Key Concepts

Law of Cosines

When solving triangles, particularly when you have two sides and the included angle (SAS situation), the Law of Cosines is your go-to tool. It allows you to find the unknown side of a triangle when you know two sides and the angle between them. The Law of Cosines states:

• \[c^2 = a^2 + b^2 - 2ab \cos(C)\]

This formula is derived from the Pythagorean theorem and it works for any kind of triangle, not just right-angled ones. In the given problem, by applying the known values in the Law of Cosines, we calculated the length of side \(c\) as approximately 4.388 miles. Understanding this formula is crucial for tackling any triangle solving problem where three elements of the triangle (either sides or angles) are known.

Law of Sines

The Law of Sines is another fundamental concept in trigonometry, used primarily for solving triangles when you have known values for angles and sides. It's especially helpful once you have one or two angles and a side, often denoted as ASA or AAS situations. It states:

• \[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]

This formula helps us find unknown angles or sides of a triangle using proportions. In our exercise, once we found all the sides with the help of the Law of Cosines, the Law of Sines helped us calculate angle \(A\) as approximately 47.12 degrees. Remember to use the inverse sine function (\(\sin^{-1}\)) to solve for an angle once you have computed its sine value.

Triangle Solving

Triangle solving involves finding unknown elements of a triangle, such as sides or angles, given a certain set of known elements. This process often requires utilizing various trigonometric laws or identities depending on the information available. For example, with the combination of side-side-angle (SAS), we initially turn to the Law of Cosines to reveal the unknown side, as in this exercise. Once that side is discovered, other tools like the Law of Sines can come into play.

• Identify the known elements: sides or angles.
• Choose the appropriate laws: Law of Sines or Cosines.
• Calculate step-by-step, checking each step for correctness.

Triangle solving enables you to fully understand the dimensions and angles of a triangle, which is essential in fields ranging from land surveying to architecture.

Angle Calculation

Angle calculation is a vital part of solving triangles. After determining the sides using trigonometric laws, you often need to find the angles to complete the triangle's dimensions. This can be done through the sum of the triangle's angles, which is always 180 degrees, and the application of trigonometric functions.

• Use the Law of Sines and your known values to find unknown angles.
• Apply inverse trigonometric functions to solve for specific angles.
• Always remember the sum of angles in a triangle is 180 degrees.

In our example, once side \(c\) was known, the angles \(A\) and \(B\) were calculated using the Law of Sines and the angle sum property of triangles. This provided angle \(A\) as approximately 47.12 degrees and angle \(B\) as approximately 73.18 degrees, culminating in a complete understanding of the triangle's dimensions.