/*! 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 51 Evaluate. \(\tan \left(\sin ^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 51

Evaluate. \(\tan \left(\sin ^{-1} 0.1\right)\)

Short Answer

Expert verified
The value is approximately 0.1005.

Step by step solution

01

- Understand the problem

The exercise requires evaluating the expression \(\tan \left( \sin^{-1} 0.1\right)\). This involves first finding the angle whose sine is 0.1, and then finding the tangent of that angle.
02

- Find the angle for \(\sin^{-1} 0.1\)

\(\sin^{-1} 0.1\) is the angle \(\theta\) such that \(\sin \theta = 0.1\). Since \(\sin \theta = 0.1\), we can consider \(\theta\) to be the angle in a right triangle where the opposite side is 0.1 and the hypotenuse is 1.
03

- Use trigonometric relationships

Recall the identity for tangent in a right triangle: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). We already know \(\sin \theta = 0.1\). To find \(\cos \theta\), use the Pythagorean identity: \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). Substituting the known value: \(\cos \theta = \sqrt{1 - (0.1)^2} = \sqrt{0.99}\).
04

- Calculate the tangent

Now, use the values obtained: \(\tan \theta = \frac{0.1}{\sqrt{0.99}}\). This simplifies the expression to: \(\tan \theta \approx 0.1005\).

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.

inverse trigonometric functions
Inverse trigonometric functions allow us to determine the angle when given the value of a trigonometric function. For example, \(\sin^{-1} 0.1\) finds the angle \(\theta\) whose sine value is 0.1. This is also known as the arcsine function. The result of \(\sin^{-1} 0.1\) is an angle, not a ratio. In this exercise, we used \(\sin^{-1} 0.1\) to find the angle and then proceeded to calculate the tangent of that angle.
Pythagorean identity
The Pythagorean identity is a fundamental relation between the sine and cosine of an angle. It states that \[ \sin^2 \theta + \cos^2 \theta = 1 \]. This identity helps us solve many trigonometric problems. In our exercise, we used the Pythagorean identity to find the value of \[\cos \theta \], given \[ \sin \theta \]. Specifically, we had \( \sin \theta = 0.1 \) and found \( \cos \theta = \sqrt{1 - (0.1)^2} = \sqrt{0.99} \). This calculation helped us determine \[ \tan \theta = \frac{0.1}{\sqrt{0.99}} \].
right triangle relationships
Right triangle relationships are essential for understanding trigonometric functions. These include the definitions of sine, cosine, and tangent as ratios of the sides of a right triangle. In the problem \[ \tan(\sin^{-1} 0.1) \], we translated \( \sin^{-1} 0.1 \) to the context of a right triangle:
  • The opposite side is 0.1.
  • The hypotenuse is 1.
By using the Pythagorean theorem, we found the adjacent side is \[ \sqrt{0.99} \]. Then, using the definition of tangent, \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{0.1}{\sqrt{0.99}} \], we arrived at our 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

Satellite Location. \(\quad\) A satellite circles the earth in such a manner that it is \(y\) miles from the equator (north or south, height from the surface not considered) \(t\) minutes after its launch, where $$Y=5000\left[\cos \frac{\pi}{45}(t-10)\right]$$ At what times \(t\) on the interval \([0,240],\) the first \(4 \mathrm{hr}\) is the satellite 3000 mi north of the equator?

Solve in \([0,2 \pi)\),. $$|\sin x|=\frac{\sqrt{3}}{2}$$

Solve, finding all solutions in \([0,2 \pi)\). $$5 \cos 2 x+\sin x=4$$,.$$

Rationalize the numerator. $$\sqrt{\frac{\cos ^{2} x}{2 \sin ^{2} x}}$$

Assuming that \(\sin \theta=0.6249\) and \(\cos \phi=0.1102\) and that both \(\theta\) and \(\phi\) are first-quadrant angles, evaluate each of the following. $$\tan (\theta+\phi)$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Discrete 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.