Chapter 6: Q. 97 (page 412)

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection. $y = \cos x y = \frac{1}{2}$

Short Answer

Expert verified

The four points of intersection are $(\frac{π}{3}, \frac{1}{2}), (\frac{5 π}{3}, \frac{1}{2}), (- \frac{π}{3}, \frac{1}{2}), (- \frac{5 π}{3}, \frac{1}{2})$ .

Step by step solution

Step 1. Given Information

We are given two equations $y = \cos x$ and $y = \frac{1}{2}$ .

We need to find any four points of intersection of the two graphs.

We will graph the two equations and then find the first point of intersection and then using the properties of cosine function we will find the other three points.

Step 2. Graph the functions

The graph of the two equations is given as

It can be seen that the line and the sinusoidal curve intersect each other at infinitely many points and the first point of intersection is $(\frac{π}{3}, \frac{1}{2})$ .

Step 2. Find the second point

As the function is cyclic, so $\cos x = \cos (2 π - x)$ .

$\cos (\frac{π}{3}) = \cos (2 π - \frac{π}{3}) \cos (\frac{π}{3}) = \cos (\frac{5 π}{3})$

Thus the second point will be $(\frac{5 π}{3}, \frac{1}{2})$

Step 4. Find the third and fourth point

As cosine function is even. So $\cos x = \cos (- x)$

Now,

$\cos (\frac{π}{3}) = \cos (- \frac{π}{3})$

and

$\cos (\frac{5 π}{3}) = \cos (- \frac{5 π}{3})$

So the third and fourth points are $(- \frac{π}{3}, \frac{1}{2}), (- \frac{5 π}{3}, \frac{1}{2})$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection. $y = \cos x y = \frac{1}{2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Graph the functions

Step 2. Find the second point

Step 4. Find the third and fourth point

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Geometry

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection. y=cosxy=12

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Graph the functions

Step 2. Find the second point

Step 4. Find the third and fourth point

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Discrete Mathematics

Geometry

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

The graphs of the given pairs of functions intersect infinitely many times. Find four of these points of intersection. $y = \cos x y = \frac{1}{2}$