/*! 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 26 Solve each equation on the inter... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 26

Solve each equation on the interval \(4 \cos ^{2} \theta-3=0\)

Short Answer

Expert verified
\( \theta = 30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ} \)

Step by step solution

01

Isolate the trigonometric function

Given the equation: \[4 \cos^2 \theta - 3 = 0\]First, add 3 to both sides to isolate the trigonometric term:\[4 \cos^2 \theta = 3\]
02

Solve for \( \cos^2 \theta \)

Divide both sides by 4:\[ \cos^2 \theta = \frac{3}{4}\]
03

Take the square root

Take the square root of both sides, remembering to consider both the positive and negative square roots:\[ \cos \theta = \pm \frac{\sqrt{3}}{2} \]
04

Determine the angles

Find the angles \( \theta \) where \( \cos \theta = \frac{\sqrt{3}}{2} \) and \( \cos \theta = -\frac{\sqrt{3}}{2} \).For \( \cos \theta = \frac{\sqrt{3}}{2} \): \( \theta = 30^{\circ} \) or \( \theta = 330^{\circ} \).For \( \cos \theta = -\frac{\sqrt{3}}{2} \): \( \theta = 150^{\circ} \) or \( \theta = 210^{\circ} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Cosine Function
The cosine function is one of the basic trigonometric functions, often abbreviated as \(\text{cos}\). It relates the angle of a right-angled triangle to the ratio of the length of the adjacent side over the hypotenuse. This function is periodic and has values which range between -1 and 1. In the unit circle, cosine represents the x-coordinate of a point.
Solving Quadratic Trigonometric Equations
Quadratic trigonometric equations, such as \[4 \cos^2 \theta - 3 = 0\], are solved similarly to regular quadratic equations. The first step is to isolate the trigonometric term. This involves rearranging the equation and possibly factoring. For example, \[4 \cos^2 \theta - 3 = 0\] simplifies to \[4 \cos^2 \theta = 3\]. Next, we solve for \[\cos^2 \theta\] by dividing both sides by 4, resulting in \[\cos^2 \theta = \frac{3}{4}\]. Taking the square root provides the solutions \[\cos \theta = \pm \frac{\text{\text\sqrt\{3\text\}}}{2}\].
Finding Solutions Within a Given Interval
When solving trigonometric equations, it is essential to consider the specified interval for the solutions. In this particular problem, we need to find all angles \(\theta\) such that the cosine of the angle equals 卤\frac{\text{\text\sqrt\{3\text\}}}{2}\ within the interval from 0掳 to 360掳. For \(\text{cos} \theta = \frac{\text{\text\sqrt\{3\text\}}}{2}\), this occurs at 30掳 and 330掳. For \(\text{cos} \theta = -\frac{\text{\text\sqrt\{3\text\}}}{2}\), the solutions are 150掳 and 210掳.
Using Trigonometric Identities Effectively
Trigonometric identities are essential tools for simplifying and solving trigonometric equations. They provide relationships between different trigonometric functions that can be used to manipulate expressions. For instance, the Pythagorean identity \[ \sin^2 \theta + \cos^2 \theta = 1 \] can be helpful in converting between sine and cosine. Familiarity with these identities allows for more agile problem-solving, especially when dealing with complex trigonometric equations.

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Most popular questions from this chapter

Find the exact value of \(\sin \frac{\pi}{3} \cos \frac{\pi}{3}\)

Show that \(\cos \left(\sin ^{-1} v+\cos ^{-1} v\right)=0\)

Area of a Dodecagon Part I A regular dodecagon is a polygon with 12 sides of equal length. See the figure. (a) The area \(A\) of a regular dodecagon is given by the formula \(A=12 r^{2} \tan \frac{\pi}{12},\) where \(r\) is the apothem, which is a line segment from the center of the polygon that is perpendicular to a side. Find the exact area of a regular dodecagon whose apothem is 10 inches. (b) The area \(A\) of a regular dodecagon is also given by the formula \(A=3 a^{2} \cot \frac{\pi}{12},\) where \(a\) is the length of a side of the polygon. Find the exact area of a regular dodecagon if the length of a side is 15 centimeters.

Challenge Problem Show that \(\tan ^{-1}\left(\frac{1}{v}\right)=\frac{\pi}{2}-\tan ^{-1} v\) if \(v>0\).

Find the exact value of each expression. $$ \tan \left(\sin ^{-1} \frac{4}{5}+\cos ^{-1} 1\right) $$
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