Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 2

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=29 \mathrm{~cm}, \mathrm{BC}=41 \mathrm{~cm}, B=100^{\circ}\)

Short Answer

Expert verified
Question: In triangle ABC, given side AB = 29 cm, side BC = 41 cm, and angle B = 100°, find the remaining components: side AC, angle A, and angle C. Solution: By applying the Law of Cosines and Law of Sines, we found the following missing components of triangle ABC: - Side AC ≈ 51.15 cm - Angle A ≈ 42.49° - Angle C ≈ 37.51°

Step by step solution

01

Identify the given information

In this problem, we are given: - Side AB = 29 cm - Side BC = 41 cm - Angle B = 100° We have the task to find the remaining components of the triangle: - Angle A - Angle C - Side AC
02

Use the Law of Cosines to find side AC

First, we will use the Law of Cosines to find the third side length, AC. Using the given data, the Law of Cosines formula becomes: \[AC^2 = AB^2 + BC^2 - 2 \cdot AB \cdot BC \cdot \cos(B)\] Substitute the values given: \[AC^2 = 29^2 + 41^2 - 2 \cdot 29 \cdot 41 \cdot \cos(100^{\circ})\] Now, we can calculate the value of AC: \[AC \approx 51.15\] So side AC is approximately 51.15 cm.
03

Use the Law of Sines to find angle A

To find angle A, we can use the Law of Sines: \[\frac{\sin{A}}{AB} = \frac{\sin{B}}{BC}\] Rearrange the formula to solve for angle A: \[\sin{A} = \frac{AB \cdot \sin{B}}{BC}\] Now, substitute the given values: \[\sin{A} = \frac{29 \cdot \sin{100^{\circ}}}{41}\] Calculate the value of sin(A) and then find the measure of angle A: \[A \approx 42.49^{\circ}\] Angle A is approximately 42.49°.
04

Use the angle sum property to find angle C

Now we can find angle C using the angle sum property of triangles, which states that the sum of the three angles in a triangle is equal to 180°. Therefore: \[A + B + C = 180^{\circ}\] Substitute the values of angle A and angle B: \[42.49^{\circ} + 100^{\circ} + C = 180^{\circ}\] Now, solve for angle C: \[C \approx 37.51^{\circ}\] So angle C is approximately 37.51°.
05

Present the final solution

Now we have found all the missing components of the triangle: - Side AC ≈ 51.15 cm - Angle A ≈ 42.49° - Angle C ≈ 37.51° Thus, we have successfully solved triangle ABC.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

In questions 1-11 \(\Delta \mathrm{ABC}\) has a right angle at \(\mathrm{C}\). Calculate AC given \(\mathrm{AB}=15 \mathrm{~cm}\) and \(\mathrm{BC}=12 \mathrm{~cm}\)

In questions \(1-10\) solve \(\triangle \mathrm{ABC}\) given \(\mathrm{AB}=42 \mathrm{~cm}, \mathrm{BC}=37 \mathrm{~cm}, \mathrm{AC}=26 \mathrm{~cm}\)

A ship travels for \(10 \mathrm{~km}\) on a bearing of \(30^{\circ} .\) It then follows a bearing of \(60^{\circ}\) for \(20 \mathrm{~km}\). Calculate the distance of the ship from the starting position.

Two vertical towers have heights \(9 \mathrm{~m}\) and \(17.2 \mathrm{~m}\) and are \(42 \mathrm{~m}\) apart. (a) Calculate the angle of elevation from the base of the shorter tower to the top of the taller tower. (b) Calculate the angle of depression from the top of the taller tower to the top of the shorter tower.

A tower has a bearing of \(37^{\circ} 00^{\prime}\) when measured from a point \(\mathrm{O}\), and is \(973 \mathrm{~m}\) distant from \(\mathrm{O}\). A chimney has a bearing of \(100^{\circ} 30^{\prime}\) when measured from \(\mathrm{O}\) and is \(1042 \mathrm{~m}\) distant from O. Calculate the distance from the tower to the chimney.
See all solutions

Recommended explanations on Physics Textbooks

Particle Model of Matter

Read Explanation

Famous Physicists

Read Explanation

Solid State Physics

Read Explanation

Electricity and Magnetism

Read Explanation

Waves Physics

Read Explanation

Nuclear Physics

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.

Sign-up for free