/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Solve the triangle, if possible.... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 7

Solve the triangle, if possible. $$c=3 \mathrm{mi}, B=37.48^{\circ}, C=32.16^{\circ}$$

Short Answer

Expert verified
A = 110.36掳, a 鈮� 5.298 mi, b 鈮� 3.441 mi

Step by step solution

01

- Find Angle A

Use the fact that the sum of the angles in a triangle is always 180掳. Thus, calculate angle A as follows: \[A = 180^\text{掳} - B - C = 180^\text{掳} - 37.48^\text{掳} - 32.16^\text{掳} = 110.36^\text{掳} \]
02

- Use the Law of Sines

We know side c and angles A, B, and C. Use the Law of Sines to find the other sides: \[ \frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)} \] Let's solve for side a first: \[ a = c \cdot \frac{\text{sin}(A)}{\text{sin}(C)} \]Substitute the known values: \[ a = 3 \cdot \frac{\text{sin}(110.36^\text{掳})}{\text{sin}(32.16^\text{掳})} \approx 3 \cdot \frac{0.936}{0.530} \approx 3 \cdot 1.766 \approx 5.298 \text{mi} \]
03

- Find side b

Use the Law of Sines again to find side b: \[ b = c \cdot \frac{\text{sin}(B)}{\text{sin}(C)} \]Substitute the known values: \[ b = 3 \cdot \frac{\text{sin}(37.48^\text{掳})}{\text{sin}(32.16^\text{掳})} \approx 3 \cdot \frac{0.608}{0.530} \approx 3 \cdot 1.147 \approx 3.441 \text{mi} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Triangle Angles Sum
The sum of all angles in a triangle is always 180 degrees. This is a fundamental property of triangles and it helps us find an unknown angle if the other two angles are known. For the given exercise, we had angles B = 37.48掳 and C = 32.16掳. To find angle A, we subtract the sum of angles B and C from 180掳:
  • Add angles B and C: 37.48掳 + 32.16掳 = 69.64掳
  • Subtract from 180掳: 180掳 - 69.64掳 = 110.36掳
So, angle A = 110.36掳. This vital step ensures that we use all given information accurately to solve the triangle.
Law of Sines
The Law of Sines is incredibly useful for solving triangles, especially when dealing with non-right triangles. The formula is \(\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}\). This ratio relation lets us find unknown sides when we know an angle and its opposite side. In the exercise, we used it to find sides a and b.
  • To find side a: \(a = c \cdot \frac{\text{sin}(A)}{\text{sin}(C)}\)
  • Substituting the known values, \(a = 3 \cdot \frac{\text{sin}(110.36掳)}{\text{sin}(32.16掳)} 鈮� 3 \cdot \frac{0.936}{0.530} 鈮� 3 \cdot 1.766 鈮� 5.298 \text{mi}\)

    • Next, we use it to find side b: \(b = c \cdot \frac{\text{sin}(B)}{\text{sin}(C)}\)
  • Substituting the known values, \(b = 3 \cdot \frac{\text{sin}(37.48掳)}{\text{sin}(32.16掳)} 鈮� 3 \cdot \frac{0.608}{0.530} 鈮� 3 \cdot 1.147 鈮� 3.441 \text{mi}\)
    • Solving Triangles
    Solving a triangle means finding all its missing sides and angles. Given certain pieces of information such as sides and angles, we use properties and laws like the Triangle Angles Sum and the Law of Sines to find the unknowns. For the given exercise:
    • We found angle A using the Triangle Angles Sum: A = 110.36掳
    • We applied the Law of Sines to find side a, getting a 鈮� 5.298 mi
    • We used the Law of Sines again to find side b, getting b 鈮� 3.441 mi
    Follow these steps in sequence, and use precise calculations. This approach ensures that each part of the puzzle falls into place, leading to a complete solution.

    One App. One Place for Learning.

    All the tools & learning materials you need for study success - in one app.

    Get started for free

    Most popular questions from this chapter

    Find the value. $$\sin 45^{\circ}$$

    Find and graph the sixth roots of \(1 .\)

    Show that for any complex numbers \(z\) and \(w\) $$ |z \cdot w|=|z| \cdot|w| $$ (Hint: Let \(z=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)\) and \(\left.w=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right) .\right)\)

    Raise the number to the given power and write standard notation for the answer. $$\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}} i\right)^{12}$$

    Find the angle between the given vectors, to the nearest tenth of a degree. $$\mathbf{u}=\langle 2,-5\rangle, \mathbf{v}=\langle 1,4\rangle$$
    See all solutions

    Recommended explanations on Math Textbooks

    Statistics

    Read Explanation

    Geometry

    Read Explanation

    Pure Maths

    Read Explanation

    Discrete Mathematics

    Read Explanation

    Applied Mathematics

    Read Explanation

    Calculus

    Read Explanation
    View all explanations

    What do you think about this solution?

    We value your feedback to improve our textbook solutions.

    Study anywhere. Anytime. Across all devices.