Problem 19 - Trigonometry

Question

$$ \text { Solve each of the following triangles. } $$ $$ a=412 \mathrm{~m}, c=342 \mathrm{~m}, B=151.5^{\circ} $$

Step-by-Step Solution

Answer

A ≈ 23.5°, B = 151.5°, C = 5°, b ≈ 566.28 m.

Step 1: Understanding the Triangle

We are given two sides and an angle that is not between them (SSA condition). Specifically, we have side \(a = 412\) m, side \(c = 342\) m, and angle \(B = 151.5^{\circ}\). We need to find angle \(A\), angle \(C\), and side \(b\).

Step 2: Use the Law of Sines

The Law of Sines states \( \frac{a}{\sin A} = \frac{c}{\sin C} = \frac{b}{\sin B}\). First, we will use this to find \(b\): \[ b = \frac{a \cdot \sin B}{\sin B} = \frac{412 \cdot \sin(151.5^{\circ})}{0.5} \approx 566.28 \ \text{m} \]

Step 3: Use Law of Sines for Angle A

Apply the Law of Sines to find \( \sin A \): \[ \sin A = \frac{a \cdot \sin B}{b} = \frac{412 \cdot \sin 151.5^{\circ}}{566.28} \approx \sin 23.5^{\circ} \] Thus, \( A \approx 23.5^{\circ} \).

Step 4: Calculate Angle C

Using the fact that the sum of angles in a triangle is \(180^{\circ}\), we can find \(C\): \[ C = 180^{\circ} - A - B = 180^{\circ} - 23.5^{\circ} - 151.5^{\circ} = 5^{\circ} \]

Step 5: Summarize the Solution

We have found all missing parts of the triangle: \(A \approx 23.5^{\circ}\), \(B = 151.5^{\circ}\), \(C = 5^{\circ}\), and \(b \approx 566.28 \ \text{m}\).

Key Concepts

Law of Sines

In trigonometry, one of the essential formulas used to solve triangles is the Law of Sines. This theorem is particularly helpful when dealing with oblique triangles, which are not right-angled. The Law of Sines states that the ratio of the length of a side to the sine of the angle opposite that side is constant for all three sides and angles of a triangle.

Mathematically, this can be expressed as:

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

This formula is invaluable, especially when you have two sides and an angle known, but it's not the angle between the sides (known as the **SSA condition**). In such cases, the Law of Sines helps us find the unknown sides or angles.

Let's use an example: if we know `side a`, `side c`, and `angle B`, we can use the Law of Sines to find the other side, `b`, and the angles, `A` and `C`. The steps involve plugging the known values into the formula, performing algebraic manipulations, and using inverse trigonometric functions to solve for the unknowns. Remember, solving triangles using this law requires careful calculation especially when determining whether a given set of values can produce a triangle.

triangle angles

Angles in a triangle are crucial as they define the shape and solve the triangle effectively. A fundamental property of triangles is that the sum of all internal angles is always \(180^{\circ}\).

This is a core principle because once you know at least two angles, you can easily find the third one. For our triangle problem, we already know one angle, \(B = 151.5^{\circ}\), and through the Law of Sines, we computed \(A \approx 23.5^{\circ}\). Calculating the third angle \(C\) becomes simple using the triangle angle sum property:

• \[C = 180^{\circ} - A - B\]

In this way, understanding triangle angles allows us to move from partially known information to a complete understanding of the triangle's geometry. Note, however, in situations like the given problem where angles are extremely acute or obtuse, double-checking calculations is vital to ensure the angles together actually sum correctly and validate the triangle's integrity.

side-angle-side (SSA) condition

The side-angle-side (SSA) condition can present a challenge when solving triangles. This condition means we know two sides of the triangle and an angle that is not directly between these sides.

With SSA, sometimes there are two possible triangles, one possible triangle, or no triangle at all. Thorough examination and cautious approach are required to solve triangles under this condition. A typical method employed is using the Law of Sines to help determine other unknown parts of the triangle.

Here's what happened in our case:

• We were given two sides: \(a = 412\) m and \(c = 342\) m, and an angle \(B = 151.5^{\circ}\).
• We initially calculate using the Law of Sines to find the corresponding unknowns: side \(b\) and angles \(A\), \(C\).
• By showing the possible calculated sides and angles fit together as a valid triangle, we confirm our solution.

Knowing that SSA doesn't guarantee a unique solution always, is crucial. It requires an understanding of the mathematical approaches and sometimes additional insights, such as considering the properties of specific angles or using alternative methods like the Law of Cosines if necessary.