/*! 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 92 If \(\theta\) is an acute angle,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 92

If \(\theta\) is an acute angle, explain why \(0<\sin \theta<1\).

Short Answer

Expert verified
For an acute angle \(\theta\), \(\text{sin }\theta = \frac{\text{opposite}}{\text{hypotenuse}}\), which is always a positive fraction less than 1.

Step by step solution

01

- Understanding the Sine Function

The sine function, \(\text{sin }\theta\), represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. For any angle \(\theta\), \(\text{sin }\theta\) is defined as \(\frac{\text{opposite}}{\text{hypotenuse}}\).
02

- Properties of an Acute Angle

An acute angle is any angle that is greater than 0 degrees but less than 90 degrees. In a right-angled triangle, the hypotenuse is always the longest side. Hence, the length of the side opposite the angle will always be less than the length of the hypotenuse.
03

- Calculating the Sine of an Acute Angle

Since the opposite side is always shorter than the hypotenuse, and both are positive lengths, the ratio \(\frac{\text{opposite}}{\text{hypotenuse}}\) will always be a positive fractional value between 0 and 1. Therefore, \(\text{sin }\theta\) for an acute angle \(\theta\) must satisfy \(\text{0 < sin }\theta < 1\).

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.

sine function
The sine function is a fundamental concept in trigonometry. When dealing with right-angled triangles, the sine of an angle \( \theta \) represents the ratio of the length of the opposite side to the hypotenuse.
This relationship can be expressed as \( \text{sin }\theta = \frac{\text{opposite}}{\text{hypotenuse}} \).
Understanding this ratio is crucial for solving problems involving angles and side lengths in triangles.
acute angles
An acute angle is any angle that is greater than 0 degrees but less than 90 degrees.
In the context of a right-angled triangle, angles smaller than 90 degrees are considered acute.
This is important because it affects the sine values.
Given that the hypotenuse is always the longest side in a right-angled triangle, the side opposite to an acute angle is always shorter.
As a result, when you take the sine of an acute angle, you'll get a value between 0 and 1.
trigonometric ratios
Trigonometric ratios are essential for understanding the relationships between the angles and sides of a triangle.
There are three primary trigonometric functions: sine, cosine, and tangent.
In our focus on the sine function for an acute angle, important points include:
  • Since the opposite side is always shorter than the hypotenuse, \( \frac{\text{opposite}}{\text{hypotenuse}} \) is always a positive fraction.
  • This results in a value of \( \text{sin }\theta \) that lies between 0 and 1 for acute angles.
  • The same principles apply to other trigonometric ratios like cosine and tangent.
Understanding these ratios helps in solving various trigonometric problems involving right-angled triangles.

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

Given \(\cot \theta=2,\) use trigonometric identities to find the exact value of (a) \(\tan \theta\) (b) \(\csc ^{2} \theta\) (c) \(\tan \left(\frac{\pi}{2}-\theta\right)\) (d) \(\sec ^{2} \theta\)

A Blu-ray drive has a maximum speed of 10,000 revolutions per minute. If a Blu-ray disc has a diameter of \(12 \mathrm{~cm},\) what is the linear speed, in \(\mathrm{km} / \mathrm{h},\) of a point \(4 \mathrm{~cm}\) from the center if the disc is spinning at a rate of 8000 revolutions per minute?

Given \(\sec \theta=3,\) use trigonometric identities to find the exact value of (a) \(\cos \theta\) (b) \(\tan ^{2} \theta\) (c) \(\csc \left(90^{\circ}-\theta\right)\) (d) \(\sin ^{2} \theta\)

Determine the amplitude and period of each function without graphing. $$ y=6 \sin (\pi x) $$

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \sec 540^{\circ} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.