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Chapter 11: Problem 10

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. \(\alpha=30^{\circ}, a=7, b=14\)

Short Answer

Expert verified
\( \beta = 90^{\circ}, \gamma = 60^{\circ}, c = 7\sqrt{3} \).

Step by step solution

01

Identify the given data

We are given \( \alpha = 30^{\circ} \), \( a = 7 \), and \( b = 14 \). Our goal is to find the angle \( \beta \), side \( c \), and angle \( \gamma \).
02

Apply the Law of Sines

According to the Law of Sines, \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \). Substitute the given values: \[ \frac{7}{\sin 30^{\circ}} = \frac{14}{\sin \beta} \].
03

Solve for \( \beta \)

Since \( \sin 30^{\circ} = \frac{1}{2} \), the equation becomes \( \frac{7}{0.5} = \frac{14}{\sin \beta} \). Simplify to get \( 14 = \frac{14}{\sin \beta} \). Thus, \( \sin \beta = 1 \) which gives \( \beta = 90^{\circ} \) since the sine of 90 degrees is 1.
04

Find the third angle

Now, use the fact that the sum of angles in a triangle is always \( 180^{\circ} \). So, \( \gamma = 180^{\circ} - \alpha - \beta = 180^{\circ} - 30^{\circ} - 90^{\circ} = 60^{\circ} \).
05

Solve for side \( c \) using the Law of Sines

Apply the Law of Sines again: \( \frac{a}{\sin \alpha} = \frac{c}{\sin \gamma} \). Substitute the known values: \[ \frac{7}{\sin 30^{\circ}} = \frac{c}{\sin 60^{\circ}} \]. \( \sin 60^{\circ} \) is \( \frac{\sqrt{3}}{2} \), so the equation becomes \( 14 = \frac{c}{\frac{\sqrt{3}}{2}} \). Solving this gives \( c = 7\sqrt{3} \).

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Key Concepts

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

Understanding Triangle Angles
Triangles are fascinating shapes with three sides and three angles. One essential property of triangles is that the sum of their interior angles is always 180 degrees. This rule helps us solve for unknown angles when we know one or two of them. For example, if we already know two angles of a triangle, we can easily find the third angle by subtracting the sum of the known angles from 180 degrees.

Consider a triangle with angles \( \alpha, \beta, \text{ and } \gamma \). If \( \alpha = 30^{\circ} \) and \( \beta = 90^{\circ} \), as given in our problem, we can determine the third angle \( \gamma \) by calculating:
  • \( \gamma = 180^{\circ} - 30^{\circ} - 90^{\circ} \)
  • This results in \( \gamma = 60^{\circ} \)
Therefore, knowing the property of triangle angles allows us to solve for unknown values when certain angles are already specified.
Exploring the Sine Function
The sine function is a crucial concept when working with triangles, particularly in trigonometry. It relates the angles of a triangle to the ratios of its sides. The sine function is particularly vital in the Law of Sines, which we use to solve problems involving triangles with given angles or sides. The sine of an angle in a right triangle is the ratio of the length of the side opposite that angle to the hypotenuse.

In our exercise, we applied the sine function to find unknown values. We are given:
  • \( \sin 30^{\circ} = 0.5 \)
  • \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)
  • \( \sin 90^{\circ} = 1 \)
These values are used to set up equations that help us determine the missing sides and angles using the Law of Sines. Understanding how to calculate and apply the sine function is essential for solving triangles.
The Process of Solving Triangles
Solving a triangle means finding all its side lengths and angle measures when some of these values are unknown but others are provided. The Law of Sines is instrumental in this process, as it connects unknown lengths or angles with known ones. This law states:

\[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]

In our example, we were given two sides and one angle. To find the remaining side and angles of the triangle, we followed these steps:
  • Identify the given values and assign them to the relevant parts of the triangle
  • Use the Law of Sines to set up equations based on these given values
  • Solve these equations to discover unknown angles, such as \( \beta = 90^{\circ} \)
  • Calculate the third angle using the triangle sum property \( \gamma = 60^{\circ} \)
  • Lastly, apply the Law of Sines again to find the unknown side \( c = 7\sqrt{3} \)
By understanding and applying these processes, solving a triangle becomes a straightforward task.

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Most popular questions from this chapter

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