/*! 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 5 Solve each right triangle in Exe... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 5

Solve each right triangle in Exercises \(1-18 .\) Remember that \(m \angle C=90^{\circ} .\) Round to the same number of decimal places as in the given information. $$a=12 \text { and } c=13$$

Short Answer

Expert verified
The missing side is \(b = 5\). The angles are \(\angle A \approx 22.62^{\text{掳}}\) and \(\angle B \approx 67.38^{\text{掳}}\).

Step by step solution

01

- Identify the given information

The given sides of the right triangle are: \(a = 12\) and \(c = 13\), where \(a\) is one of the legs, and \(c\) is the hypotenuse.
02

- Use the Pythagorean Theorem

Apply the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] Plug in the given values: \[ 12^2 + b^2 = 13^2 \] \[ 144 + b^2 = 169 \]
03

- Solve for the missing side \(b\)

Rearrange the equation to isolate \(b^2\): \[ b^2 = 169 - 144 \] \[ b^2 = 25 \] Take the square root of both sides: \( b = 5 \)
04

- Calculate the angles using trigonometric ratios

Use the definition of the cosine function to find \(\angle A\): \[ \cos(A) = \frac{a}{c} = \frac{12}{13} \] Find \(\angle A\) using the inverse cosine function: \[ A = \cos^{-1}(\frac{12}{13}) \approx 22.62^{\text{掳}} \] Since \(m \angle C = 90^{\text{掳}} \), \(\angle B\) can be found as: \[ B = 90^{\text{掳}} - A \approx 90^{\text{掳}} - 22.62^{\text{掳}} \approx 67.38^{\text{掳}} \]

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.

Pythagorean Theorem
The Pythagorean Theorem is a fundamental tool in geometry, especially for right triangles. It states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. This can be written as: \[ a^2 + b^2 = c^2 \].

Here, \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.
In the given problem, \(a = 12\) and \(c = 13\).
We rearranged the Pythagorean Theorem to solve for \(b\):
\[ b^2 = c^2 - a^2 \]
Plugging in the values, we found: \[ b^2 = 169 - 144 = 25 \]
And then taking the square root of both sides, we found \(b = 5\).

Useful tip: Always make sure to double-check your calculations for accuracy.
Trigonometric Ratios
Trigonometric ratios are ratios of two sides of a right triangle. There are three primary trigonometric ratios: sine, cosine, and tangent. These ratios help in finding missing angles or sides of right triangles.
For angle \(A\) in the given triangle:
  • Cosine (cos): \( \text{cos}(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c} \)
  • Sine (sin): \( \text{sin}(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c} \)
  • Tangent (tan): \( \text{tan}(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a} \)
Using the given values, \(\text{cos}(A) = \frac{12}{13}\), which is found by the ratio of the length of the adjacent side to the hypotenuse.
Inverse Cosine
The inverse cosine function, also known as \(\text{cos}^{-1}\), allows us to find an angle when we know the cosine of that angle. It's a powerful function for solving right triangles.
To find angle \(A\) in our problem, we used the inverse cosine function:\( A = \text{cos}^{-1}\bigg(\frac{12}{13}\bigg) \approx 22.62^\text{掳} \).

Remember that a right triangle's angles always add up to \(90^\text{掳}\).
So, to find the remaining angle \(B\), we calculated:
\( B = 90^\text{掳} - 22.62^\text{掳} \approx 67.38^\text{掳} \).

Note: Always ensure your calculator is set to degree mode when working with angles in degrees!

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 a calculator to find each ratio correct to four decimal places. $$\tan 38^{\circ}$$

Use a calculator to find the measure of \(\angle A,\) to the nearest tenth of a degree. $$\tan A=2.5053$$

Use a calculator to find each ratio correct to four decimal places. $$\tan 81.43^{\circ}$$

Use your calculator to find each trigonometric ratio, correct to four decimal places. $$\cos 5^{\circ}$$

Solve each right triangle in Exercises \(1-18 .\) Remember that \(m \angle C=90^{\circ} .\) Round to the same number of decimal places as in the given information. $$c=29 \text { and } m \angle B=55^{\circ}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical 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.