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Chapter 9: Problem 5

Solve the triangles with the given parts. $$c=4380, A=37.4^{\circ}, B=34.6^{\circ}$$

Short Answer

Expert verified
The sides are \( a \approx 2812.35 \) and \( b \approx 2652.34 \); angle C is 108掳.

Step by step solution

01

Determine Angle C

To find angle C in the triangle, use the fact that the sum of the angles in any triangle is 180掳. Therefore, angle C can be calculated as follows: \[ C = 180掳 - A - B = 180掳 - 37.4掳 - 34.6掳. \] Thus, \[ C = 108掳. \]
02

Use the Law of Sines to Find Side a

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We'll use this to find side \( a \):\[ \frac{a}{\sin 37.4掳} = \frac{4380}{\sin 108掳}. \]Solving for \( a \), we get \[ a = 4380 \times \frac{\sin 37.4掳}{\sin 108掳}. \]After calculating, \( a \approx 2812.35 \) units.
03

Use the Law of Sines to Find Side b

Apply the Law of Sines again to solve for side \( b \):\[ \frac{b}{\sin 34.6掳} = \frac{4380}{\sin 108掳}. \]Therefore, \( b = 4380 \times \frac{\sin 34.6掳}{\sin 108掳}. \)After performing the calculations, \( b \approx 2652.34 \) units.

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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, especially for solving triangles when given either two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA). This law states:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]This means that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
  • To use this law effectively, you should know at least one pair of a side and its opposite angle.
  • In our example, we start with side \( c = 4380 \) and angle \( C = 108^\circ \). We used the other known angles, \( A = 37.4^\circ \) and \( B = 34.6^\circ \), to solve for sides \( a \) and \( b \).
  • The Law of Sines helps us convert angle information into length information based on these angle-side pair relationships by rearranging the formula to solve for unknown sides.
With this technique, you can find unknown sides effortlessly once you understand the relationships between the angles and their opposite sides.
Angle Sum Property
The Angle Sum Property is a foundational concept in triangle geometry. It tells us that the sum of the interior angles in a triangle is always 180 degrees.
  • To apply this in triangle solving: If you are given two angles, you can easily find the third by subtracting the sum of the given angles from 180掳.
  • In this exercise, we have \( A = 37.4^\circ \) and \( B = 34.6^\circ \). By inserting these values into the formula: \[ C = 180^\circ - A - B \]We get:\[ C = 180^\circ - 37.4^\circ - 34.6^\circ = 108^\circ \]
  • This property is not just useful in calculations but also serves as a checkpoint to ensure that your solved angles add correctly.
Understanding the Angle Sum Property helps demystify the nature of triangles and ensures your work on angle-related problems is accurate.
Triangle Solving
Triangle solving involves finding unknown parts of a triangle when given some initial information. Often, we are tasked with finding missing sides and angles using a mix of geometric laws and properties.
  • Start with what you know. Identify given sides and angles. This helps determine which mathematical principles you can use.
  • Using the Angle Sum Property, determine any unknown angle. As in our case, with two angles known, it was straightforward to find angle \( C \).
  • Apply the Law of Sines or Cosines as needed. Since we had angle-side pairs, the Law of Sines was ideal for finding the unknown sides \( a \) and \( b \).
  • Perform all calculations carefully, most importantly, solve step-by-step to avoid errors.
  • Finally, verify your answers. Ensure that the triangle's internal angle sum is 180掳 and the solved proportions align with the Law of Sines.
Triangle solving is an incredibly systematic process. By understanding the relationships between angles and sides, you can solve any triangle configuration effectively.

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Most popular questions from this chapter

In Exercises \(15-30,\) add the given vectors by using the trigonometric functions and the Pythagorean theorem. $$\begin{aligned} &E=1653, \theta_{E}=36.37^{\circ}\\\ &F=9807, \theta_{F}=253.06^{\circ} \end{aligned}$$

Find the required horizontal and vertical components of the given vectors. With the sun directly overhead, a plane is taking off at \(125 \mathrm{km} / \mathrm{h}\) at an angle of \(22.0^{\circ}\) above the horizontal. How fast is the plane's shadow moving along the runway?

Solve the triangles with the given parts. $$a=385.4, b=467.7, c=800.9$$

In Exercises \(7-14,\) with the given sets of components, find \(R\) and \(\theta\). $$R_{x}=-729, R_{y}=-209$$

Use the law of sines to solve the given problems. When an airplane is landing at an 8250 -ft runway, the angles of depression to the ends of the runway are \(10.0^{\circ}\) and \(13.5^{\circ} .\) How far is the plane from the near end of the runway?
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