Find the point of intersection

Question

of y = 2x - 4 AND y = 2cos^2π/₃x - 4

 

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I can't seem to solve this! Halp!!!

Answer 1

Hello Bill,

 

If this is a problem from a Trigonometry course, then here's the only way I could interpret your question:

 

     Find the point of intersection between:

          y = 2x - 4  and

          y = 2cos(2πx/3) - 4.

 

Any other interpretation of the cosine term either has no solution or is beyond the scope of your course.

 

Going forward with this, if you equate the two equations, you'll end up with:

 

     2x - 4 = 2cos(2πx/3) - 4

 

which will simplify with a bit of addition and division, to:

 

     x = cos(2πx/3).

 

As it stands, you won't be able to solve this for x by hand. At this point, I have to make a giant leap of faith concerning your specific instructor and your class level.

 

Normally, the following trick isn't taught until the undergraduate level in college, but I haven't taken a Trigonometry course in many years... so maybe they're teaching it in Trig by now.  :)

 

Here's the trick:

 

     cos(u) ≈ 1 - u²/2 + u⁴/24 - u⁶/720 + (more terms follow)

     ...as long as u is pretty small.

 

Now, the trick works extremely well, as long as you include enough terms on the right-hand side of the equation. Also, as u gets larger, you need to include more and more terms to stay very accurate.

 

Fortunately for this problem, you can get away with only using the first two terms:

 

     cos(u) ≈ 1 - u²/2

 

We could make an argument for adding the third term to be extremely accurate, but if we do, we'll be dealing with an equation with a fourth power of u (called a quartic equation), but solving quartic equations by hand is a subject for upper-division math majors in college. So, we'll just go with the first two terms. Your answer will be accurate to the first decimal digit.

 

So! Let's make the substitution:

 

     x = cos(2πx/3)

 

     x = 1 - (2πx/3)²/2

 

Do some manipulation, and you'll end up with:

 

     (2π²/9)x² + x - 1 = 0

 

And we can solve this for x using the quadratic equation:

 

     x = (1/2a)(-b ± sqrt(b² - 4ac))

 

Choose the value for x that is positive (the negative answer doesn't actually intersect the original cosine curve), and find the matching value for y.

 

After all of that work, you're done! Be sure not to include more than one digit after the decimal point in your answer. Round off.

 

Hope this helped! :)

Answer 2

Set the functions equal to each other.

 

2x - 4 = cos(2π/3)x - 4

 

 

Then you would reduce this equation to

 

2x = cos(2π/3)x

 

 

Now solve for x.