/*! 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 17 In \(11-22,\) solve each triangl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 17

In \(11-22,\) solve each triangle, that is, find the measures of the remaining three parts of the triangle to the nearest integer or the nearest degree. In \(\triangle R S T, r=38, s=28,\) and \(t=18\)

Short Answer

Expert verified
Angles: 109掳, 44掳, 27掳.

Step by step solution

01

Identify Known Parts and Use the Law of Cosines

We are given the side lengths of triangle \( \triangle RST \): \( r = 38 \), \( s = 28 \), and \( t = 18 \). The Law of Cosines helps us find angles when all sides are known. To find angle \( R \), we use the formula: \[ \cos R = \frac{s^2 + t^2 - r^2}{2st} \].
02

Calculate Angle R

Substitute the known values into the formula: \[ \cos R = \frac{28^2 + 18^2 - 38^2}{2 \times 28 \times 18} \]This simplifies to:\[ \cos R = \frac{784 + 324 - 1444}{1008} = \frac{-336}{1008} \approx -0.3333 \]Therefore, \( R \approx \cos^{-1}(-0.3333) \approx 109^\circ \).
03

Use the Law of Sines to Find Angle S

With angle \( R \) known, use the Law of Sines:\[ \frac{r}{\sin R} = \frac{s}{\sin S} \]Substitute the known values and solve for \( \sin S \):\[ \frac{38}{\sin 109^\circ} = \frac{28}{\sin S} \]Calculate \( \sin 109^\circ \), then solve for \( \sin S \):\[ \sin S = \frac{28 \times \sin 109^\circ}{38} \]\( \sin S \approx \frac{28 \times 0.9455}{38} \approx 0.696 \)Use \( \sin^{-1}(0.696) \) to find \( S \approx 44^\circ \).
04

Find Angle T

Now use the fact that the sum of angles in a triangle is 180掳:\[ T = 180^\circ - R - S \]Substitute \( R = 109^\circ \) and \( S = 44^\circ \):\[ T = 180^\circ - 109^\circ - 44^\circ = 27^\circ \].
05

Conclusion

The measures of the angles in \( \triangle RST \) are \( R = 109^\circ \), \( S = 44^\circ \), and \( T = 27^\circ \).

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 Solving
Solving a triangle involves finding the measurements of all its angles and sides. To do this systematically, we start by gathering the information available and using trigonometric laws to find unknown values.
  • In the case of triangle \( \triangle RST \), we began with the side lengths \( r = 38 \), \( s = 28 \), and \( t = 18 \).
  • Given these sides, the next task was to find the angles of the triangle. By determining all parts, we successfully solve the entire triangle.
  • The goal is to ensure each unknown side or angle adheres to the fundamental trigonometric principles in a triangle, such as the sum of angles being \( 180^\circ \).
Taking a strategic approach, we first used the Law of Cosines to determine one angle, then applied supplementary trigonometric rules like the Law of Sines and the fundamental angle sum property to find the remaining angles.
Law of Sines
The Law of Sines is a powerful tool for solving triangles, especially when we know an angle and its opposite side. It enables us to find unknown angles or sides of the triangle.
  • It relates the ratio of each side to the sine of its opposite angle.
  • In mathematical terms, the Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \), where \( a, b, \) and \( c \) are the sides, and \( A, B, \) and \( C \) are the opposite angles.
When applied to \( \triangle RST \), after finding angle \( R \) using the Law of Cosines, we knew \( R \), so we used the Law of Sines to find angle \( S \). By setting up the equation \( \frac{38}{\sin 109^\circ} = \frac{28}{\sin S} \), we isolated \( \sin S \) to find the angle. Such application is stepwise and intuitive, showcasing the power of this method.
Angle Calculation
Calculating angles in triangles is a key skill in geometry and trigonometry. Once you know some elements of the triangle, you can use various rules to find the unknown angles.
  • For triangle \( \triangle RST \), once angle \( R \) was determined, angle \( S \) was easily calculated using the Law of Sines.
  • Angle calculation then progresses naturally; finding one angle simplifies finding the others through subtraction and known formulae.
  • This step-by-step process ensures accuracy and reliance on the geometric properties of triangles.
Remember, finding angles in a triangle is manageable once one angle is known, as shown in the remaining calculation for angle \( T \): using \( 180^\circ - R - S \), relying on the fundamental triangle property.
Trigonometry
Trigonometry is more than just finding angles and lengths; it is about understanding the relationships within the triangle. This mathematical field uses specific functions to express these relationships and solve complex geometrical problems.
  • The core functions used include sine, cosine, and tangent, each representing a ratio of triangle sides.
  • For solving triangle \( \triangle RST \), both the Law of Cosines and the Law of Sines demonstrated how these trigonometric functions underpin our ability to solve the triangle fully.
  • We dealt with expressions such as \( \cos R \) for finding the angle initially, then turning to \( \sin S \) to further unlock more components of the triangle.
Ultimately, trigonometry reveals the systematic elegance of triangles, using formulas and relationships that enable anyone with basic trig knowledge to solve even the complex sides of geometrical puzzles.

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

Explain why an angle of depression is always congruent to an angle of elevation.

A ladder that is 15 feet long is placed so that it reaches from level ground to the top of a vertical wall that is 13 feet high. a. Use the Law of Sines to find the angle that the ladder makes with the ground to the nearest hundredth. b. Is more than one position of the ladder possible? Explain your answer.

A pole is braced by two wires that extend from the top of the pole to the ground. The lengths of the wires are 16 feet and 18 feet and the measure of the angle between the wires is \(110^{\circ} .\) Find, to the nearest foot, the distance between the points at which the wires are fastened to the ground.

From point \(C\) at the top of a cliff, two points, \(A\) and \(B,\) are sited on level ground. Points \(A\) and \(B\) are on a straight line with \(D,\) a point directly below \(C .\) The angle of depression of the nearer point, \(A,\) is 72 degrees and the angle of depression of the farther point, \(B,\) is 48 degree. If the points \(A\) and \(B\) are 20 feet apart, what is the height of the cliff to the nearest foot?

In \(13-18,\) find, to the nearest degree, the measure of each angle of the triangle with the given measures of the sides. $$ 12,20,22 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

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.