Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(A=87.2^{\circ}, b=75.9\) yards, \(C=74.3^{\circ}\)

Step-by-Step Solution

Answer

Angle \(B = 18.5^\circ\), sides \(a \approx 238.6\) yards, \(c \approx 229.9\) yards.

Step 1: Understand the Triangle's Elements

We are given the triangle with angles \(A\) and \(C\), and a side \(b\). The goal is to find the third angle \(B\), and the remaining sides \(a\) and \(c\).

Step 2: Find the Missing Angle

To find angle \(B\), remember that the sum of angles in a triangle is always \(180^\circ\). Thus, we calculate:\[B = 180^\circ - A - C = 180^\circ - 87.2^\circ - 74.3^\circ = 18.5^\circ\]

Step 3: Use the Law of Sines to Find Side a

The Law of Sines states \(\frac{a}{\sin A} = \frac{b}{\sin B}\). First plug in the known values to solve for \(a\):\[\frac{a}{\sin 87.2^\circ} = \frac{75.9}{\sin 18.5^\circ}\]Calculate \(a\):\[a = \frac{75.9 \times \sin 87.2^\circ}{\sin 18.5^\circ}\]Using a calculator to find the sines, then calculate:\[a \approx \frac{75.9 \times 0.9992}{0.3173} \approx 238.6 \text{ yards}\]

Step 4: Use the Law of Sines to Find Side c

Now use the Law of Sines to find \(c\) as follows:\[\frac{c}{\sin C} = \frac{b}{\sin B}\]Plug in the known values:\[\frac{c}{\sin 74.3^\circ} = \frac{75.9}{\sin 18.5^\circ}\]Calculate \(c\):\[c = \frac{75.9 \times \sin 74.3^\circ}{\sin 18.5^\circ}\]Using a calculator to find the sines, then calculate:\[c \approx \frac{75.9 \times 0.9613}{0.3173} \approx 229.9 \text{ yards}\]

Key Concepts

Law of Sines

The Law of Sines is a powerful tool in trigonometry, especially when solving triangles, as it relates the angles and sides in a simple proportionality. It asserts that in any triangle:

• The ratio of the length of a side to the sine of its opposite angle is the same for all three sides.

This can be expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This means if we know two angles and a side, we can find the unknown sides. In our exercise, with angles \(A\) and \(C\) and side \(b\) given, we leveraged the Law of Sines to find sides \(a\) and \(c\). The calculations involved solving for \(a\) by using:\[a = \frac{b \times \sin A}{\sin B}\]Similarly, we solved for \(c\) using:\[c = \frac{b \times \sin C}{\sin B}\]Careful calculation, usually with a calculator, will then give you the lengths of the sides needed.

Angle Sum Property

The Angle Sum Property is something you might use before diving into trigonometry. It simply states that the sum of all interior angles in a triangle is always equal to \(180^\circ\). Knowing this fact allows us to find an unknown angle when the other two are given.

• For any triangle with angles \(A\), \(B\), and \(C\), the Angle Sum Property gives us:

\[A + B + C = 180^\circ\]In the exercise, we were given angles \(A = 87.2^\circ\) and \(C = 74.3^\circ\). Using the Angle Sum Property, we calculated angle \(B\) by:\[B = 180^\circ - A - C\]Put simply, we subtract the known angles from \(180^\circ\) to solve for \(B\), which turned out to be \(18.5^\circ\). This is an essential early step in solving for the triangle's properties.

Trigonometry

Trigonometry deals with the study of triangles, making it a key area of mathematics when you want to solve triangles. It especially focuses on the relationships between angles and sides of right-angled triangles using trigonometric functions such as sine, cosine, and tangent.

• These functions help relate the angles and lengths of sides, essential for calculating unknown values in triangles.

In scenarios with general triangles, like the one in our problem, even without a right angle, trigonometric rules such as the Law of Sines still apply.

• This makes it possible to resolve these angles and sides into manageable calculations.

Mastering these concepts allows for accurate geometric representations and is essential when working with problems involving real-world architectural, engineering, and various calculations.

Solving Triangles

Solving a triangle means finding the unknown angles and sides when certain other elements of the triangle are known. To effectively solve a triangle:

• Use known angles, sides, and properties like the Law of Sines and Angle Sum Property.
• Calculate step-by-step, ensuring each calculation's accuracy.

In our example, we utilized given data (angles \(A\) and \(C\) and side \(b\)) combined with trigonometric principles to uncover the missing pieces of the puzzle. Numerical accuracy is crucial in every step, especially with trigonometric functions that often require a scientific calculator for precision.

• By doing so, we confidently determined the unknown side lengths and angles, successfully "solving" the triangle.

Whether for academic exercises or practical applications, solving triangles equips you with the skills to tackle broader and more complex geometric problems.