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Chapter 9: Problem 7

Solve the triangles with the given parts. $$a=4.601, b=3.107, A=18.23^{\circ}$$

Short Answer

Expert verified
The triangle has angles \(A = 18.23^{\circ}\), \(B = 12.19^{\circ}\), \(C = 149.58^{\circ}\) and sides \(a = 4.601\), \(b = 3.107\), \(c = 8.569\).

Step by step solution

01

Identify Triangle Data

We're given two sides and an angle: side \(a = 4.601\), side \(b = 3.107\), and angle \(A = 18.23^{\circ}\). Since two sides and a non-included angle are known, we'll use the Law of Sines to find the remaining parts.
02

Apply the Law of Sines

According to the Law of Sines, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). We'll first find angle \(B\) by using the equation \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Substitute the known values: \[ \frac{4.601}{\sin 18.23^{\circ}} = \frac{3.107}{\sin B} \]Calculate \(\sin 18.23^{\circ}\) to solve for \(\sin B\).
03

Solve for \(\sin B\)

Calculate \(\sin 18.23^{\circ}\) using a calculator:\(\sin 18.23^{\circ} \approx 0.312\).Multiply both sides by \(\sin B\) and divide by \(4.601\):\[ \sin B = \frac{3.107 \times 0.312}{4.601} \approx 0.210 \]Now, find angle \(B\) by taking the inverse sine:\( B = \sin^{-1}(0.210) \approx 12.19^{\circ}\).
04

Calculate Angle C

With angles \(A\) and \(B\) known, calculate angle \(C\). Since the angles of a triangle sum up to \(180^{\circ}\), we have:\[ C = 180^{\circ} - 18.23^{\circ} - 12.19^{\circ} \approx 149.58^{\circ} \]
05

Use Law of Sines Again to Find Side c

Now that we have angle \(C\), use the Law of Sines again to find side \(c\):\[ \frac{c}{\sin C} = \frac{a}{\sin A} \]Substitute the known values and calculate \(\sin 149.58^{\circ}\):\[ c = \frac{4.601 \times \sin 149.58^{\circ}}{0.312} \approx 8.569 \]
06

Verify the Solution

Confirm the calculations: angles \(A\), \(B\), and \(C\) should sum to \(180^{\circ}\) and all sides and angles should satisfy the Law of Sines.The calculated values satisfy these conditions, verifying the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Law of Sines
The Law of Sines is a powerful tool in trigonometry used to solve triangles, especially when you have two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). This law states:
  • For a triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
When solving triangles like in the given exercise, we identify known sides and angles and utilize this equation to find unknown parts. Such a clear correlation between angles and their opposite sides makes it easier to find unknown components once we know some starting values.
In practice, calculate one unknown angle first and ensure that triangle properties and trigonometric rules are honored. Double-check solutions as shown in the exercise to ensure accuracy.
triangle solving
Solving a triangle means finding all its unknown sides and angles. In a problem like this, where you are given two sides and a non-included angle, it often involves several strategies:
  • First, identify what you know, and decide which rule to apply. The Law of Sines is particularly useful in this SSA (Side-Side-Angle) scenario.
  • Use the known information to find unknown angles or sides step-by-step. It's often helpful to work systematically through the options.
The step-by-step solution shows applying the Law of Sines to find angle \(B\). Once angle \(B\) is known, angle \(C\) is straightforward using the fact that the angles in a triangle sum to \(180^{\circ}\). After the angles are discovered, you can apply again the Law of Sines to find unknown side lengths.
This structured approach simplifies complex problems and ensures all triangle properties are respected.
inverse sine
Inverse sine, denoted \(\sin^{-1}\) or arcsin, is used to find an angle when you know its sine. This concept is crucial in triangle solving exercises such as this one.
When trying to find an angle given a sine value:
  • Use \(\sin^{-1}(x)\) to find an angle whose sine is \(x\).
  • This function will give you an angle in the range from \(-90^{\circ}\) to \(90^{\circ}\) for real numbers within \(-1\) to \(1\).
In the example provided, after solving for \(\sin B\), the inverse sine was used to determine \(B\). Handling angles with inverse sine requires attention to ensuring the resulting value logically fits the triangle's context.
This process makes it a key tool in verifying and solving for unknown angles in triangle problems where the use of trigonometric ratios is required.

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Most popular questions from this chapter

Add the given vectors by components. $$\begin{array}{l} A=30.7, \theta_{A}=18.2^{\circ} \\ B=45.2, \theta_{B}=251.0^{\circ} \end{array}$$

Use the law of cosines to solve the given problems. Write the form of the law of cosines, given sides \(a\) and \(b,\) and angle \(C=90^{\circ}\).

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With the given sets of components, find \(R\) and \(\theta\). $$R_{x}=6941, R_{y}=-1246$$
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