/*! 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 10 Given the indicated parts of tri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 10

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) approximate the remaining parts. $$\beta=64^{\circ} 20^{\prime}, \quad a=20.1$$

Short Answer

Expert verified
\(\alpha = 25^{\circ} 40^{\prime}\); \(b \approx 9.14\); \(c \approx 22.38\).

Step by step solution

01

Identify Known Values

In triangle \(ABC\), we know that \(\gamma = 90^{\circ}\) (since it is a right triangle), \(\beta = 64^{\circ} 20^{\prime}\), and the side \(a = 20.1\). We need to find the other angle \(\alpha\), and the remaining sides \(b\) and \(c\).
02

Calculate Angle \(\alpha\)

Since the sum of angles in any triangle is \(180^{\circ}\), we can find \(\alpha\) by using the equation: \[\alpha = 180^{\circ} - \beta - \gamma.\]Substitute the known values:\[\alpha = 180^{\circ} - 64^{\circ} 20^{\prime} - 90^{\circ} = 25^{\circ} 40^{\prime}.\]
03

Calculate Side \(b\) Using Trigonometry

Using the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle, we have:\[\tan(\beta) = \frac{a}{b}.\]Solving for \(b\):\[b = \frac{a}{\tan(\beta)}.\]Substitute \(a = 20.1\) and \(\beta = 64^{\circ} 20^{\prime}\):\[b \approx \frac{20.1}{\tan(64^{\circ} 20^{\prime})} \approx 9.14.\]
04

Calculate Hypotenuse \(c\) Using Trigonometry

We use the sine function, which is the ratio of the opposite side to the hypotenuse:\[\sin(\beta) = \frac{a}{c}.\]Solving for \(c\):\[c = \frac{a}{\sin(\beta)}.\]Substitute \(a = 20.1\) and \(\beta = 64^{\circ} 20^{\prime}\):\[c \approx \frac{20.1}{\sin(64^{\circ} 20^{\prime})} \approx 22.38.\]
05

Verification Step: Check Calculations

Double-check calculations by computing the cosine of \(\beta\) and verifying side \(b\) using:\[\cos(\beta) = \frac{b}{c}.\]Calculate \(\cos(64^{\circ} 20^{\prime})\) and compare \(b\):\[\cos(64^{\circ} 20^{\prime}) \approx 0.435\quad \Rightarrow \quad b \approx c \times 0.435 \approx 9.14.\]The values are consistent, ensuring calculations are correct.

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.

Trigonometric Functions
Trigonometric functions are fundamental tools in understanding right triangles. When dealing with a right triangle, the main functions used are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions help relate the angles to the sides of the triangle.
  • Sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse.
  • Cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
  • Tangent of an angle is the ratio of the length of the side opposite the angle to the adjacent side.
These functions make it easy to solve for unknown sides and angles in a triangle once some parts are known. In our exercise, from \(\beta = 64^{\circ} 20^{\prime}\), the tangent function was used to find side \(b\) and the sine function was used to find the hypotenuse \(c\). These calculations are straightforward when you substitute known values into the function equations.
Angle Sum Property
The angle sum property is a fundamental concept in trigonometry and geometry. It states that the sum of the angles in any triangle is always \(180^{\circ}\). This is particularly helpful in right triangles where one angle is always \(90^{\circ}\).
To find an unknown angle in a triangle, simply subtract the known angles from \(180^{\circ}\). For example, if you have a right triangle with the angles \(\gamma = 90^{\circ}\) and \(\beta = 64^{\circ} 20^{\prime}\), you can find the third angle \(\alpha\) by calculating:\[\alpha = 180^{\circ} - 90^{\circ} - 64^{\circ} 20^{\prime}\]which equals \(25^{\circ} 40^{\prime}\). This property ensures you can always determine one unknown angle if the other two are known.
Triangle Side Lengths
Understanding side lengths in a right triangle involves using trigonometric relationships. With a right triangle, we have the hypotenuse, opposite side, and adjacent side. Depending on the known values, you can calculate the missing sides using trigonometric functions like sine, cosine, and tangent.
  • Use the tangent function when you know an angle and one side, solving for the opposite or adjacent side.
  • Employ the sine function to find the hypotenuse or opposite when you have an angle and another side.
In the example provided, side \(b\) was calculated using the tangent function:\[b = \frac{a}{\tan(\beta)}\]and hypotenuse \(c\) was calculated using the sine function:\[c = \frac{a}{\sin(\beta)}\]These formulas help break down complex problems into smaller, easier-to-solve parts, giving you clear methods to find any unknown side in a right triangle.

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

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\csc \theta=5 \text { and } \cot \theta<0$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\cot \theta=\frac{3}{4} \text { and } \cos \theta<0$$

Graph the function \(f\) in the viewing rectangle \([-2 \pi, 2 \pi, \pi / 2]\) by \([-4,4] .\) Use the graph of \(f\) to predict the graph of \(g .\) Verify your prediction by graphing \(g\) in the same vlewing rectangle. $$f(x)=0.5 \csc 0.5 x, g(x)=0.5 \csc 0.5 x-2$$

Use fundamental identities to find the values of the trigonometric functions for the given conditions. $$\cos \theta=\frac{1}{2} \text { and } \sin \theta<0$$

Refer to the graph of \(y=\sin x\) or \(y=\cos x\) to find the exact values of \(x\) in the interval \([0,4 \pi]\) that satisfy the equation. $$\cos x=-1$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

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.