Problem 94 Suppose $\theta$ is an angle s... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 5: Problem 94

Suppose $\theta$ is an angle such that $\cos \theta$ is rational. Explain why $\cos (2 \theta)$ is rational.

Short Answer

Expert verified

Recall the double angle formula for cosine: $\cos (2\theta) = 2 \cos^2 (\theta) - 1$. Given $\cos \theta$ is rational, let $\cos \theta = \frac{a}{b}$, with $a$ and $b$ as integers and $b\neq0$. Substituting this value, we get $\cos (2\theta) = 2 \cdot \frac{a^2}{b^2} - 1$. Rewrite this expression as a rational number: $\cos (2\theta) = \frac{2a^2 - b^2}{b^2}$. Since both the numerator and denominator are integers, $\cos (2\theta)$ is rational.

Step by step solution

Recall the double angle formula for cosine

The double angle formula for cosine is given by: \[ \cos (2\theta) = 2 \cos^2 (\theta) - 1 \] This formula will help us find $\cos (2\theta)$ using $\cos \theta$, which is given to be a rational number.

Substitute the given rational value of $\cos \theta$

Since $\cos \theta$ is a rational number, let's say that $\cos \theta = \frac{a}{b}$, where $a$ and $b$ are integers and $b\neq0$. Using the double angle formula, we'll substitute this value in the expression of $\cos (2\theta)$: \[ \cos (2\theta) = 2 \left(\frac{a}{b}\right)^2 - 1\]

Simplify the expression

Now, let's simplify the expression: \[ \cos (2\theta) = 2 \cdot \frac{a^2}{b^2} - 1 \]

Make the expression look like a rational number

We need to rewrite the expression in such a way that it looks like a rational number, i.e., in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\neq0$. To do that, we can rewrite the above expression as: \[ \cos (2\theta) = \frac{2a^2}{b^2} - \frac{b^2}{b^2}\]

Combine the two fractions

Next, combine the two fractions: \[ \cos (2\theta) = \frac{2a^2 - b^2}{b^2} \] Given that $2a^2 - b^2$ is an integer and $b^2$ is an integer ($b\neq0$), we can now conclude that $\cos (2\theta)$ is a rational number.

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!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Suppose \(\theta\) is an angle such that \(\cos \theta\) is rational. Explain why \(\cos (2 \theta)\) is rational.

Short Answer

Step by step solution

Recall the double angle formula for cosine

Substitute the given rational value of \(\cos \theta\)

Simplify the expression

Make the expression look like a rational number

Combine the two fractions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Pure Maths

Decision Maths

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.