Finding sum of roots of trigonometric equation

In summary, the sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is: (b) 4950 π.
  • #1
Priyadarshi Raj
8
1

Homework Statement


Question:
Sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is:
(a) 5050 π
(b) 4950 π
(c) 5151 π
(d) none of these

The correct answer is: (b) 4950 π

Homework Equations


## cos(2x) = 2cos^2(x) -1 ##

The Attempt at a Solution


##tan^2 (33x) = cos(2x)-1##
⇒ ##tan^2 (33x) = 2cos^2(x)-2##
⇒ ##\frac{sin^2 (33x)}{cos^2(33x)} = 2(cos^2 (x) -1)##
⇒ ##sin^2 (33x) = 2cos^2(x)cos^2(33x) - 2cos^2 (33x)##
⇒ ##1 - cos^2(33x)= 2cos^2(x)cos^2(33x) - 2cos^2 (33x)##
⇒ ##2cos^2(x)cos^2(33x) - cos^2 (33x) - 1 = 0##

And I end up making it more complex. It'b be great if I could convert the ##33x## to something smaller.

Please help.
 
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  • #2
The trick here, I believe, is to realize that reduction of the [itex]33x[/itex] is probably not a feasible option at all - so what you want to get in the end is not a quadratic or polynomial equation, but a simple product of trigonometric functions being equal to a nice number.

So the hint is to use the trigonometric identity [itex]\sec^2 33x = 1 + \tan^2 33x[/itex] and keep the [itex]\cos 2x[/itex] term as it is.

Edit: I realized a much much simpler method. Note that the LHS is [itex]\geq 0[/itex]. How about the RHS? This gives you the answer rightaway.
 
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Likes Samy_A and member 587159
  • #3
Priyadarshi Raj said:

Homework Statement


Question:
Sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is:
(a) 5050 π
(b) 4950 π
(c) 5151 π
(d) none of these

The correct answer is: (b) 4950 π

Homework Equations


## cos(2x) = 2cos^2(x) -1 ##

The Attempt at a Solution


##tan^2 (33x) = cos(2x)-1##
⇒ ##tan^2 (33x) = 2cos^2(x)-2##
⇒ ##\frac{sin^2 (33x)}{cos^2(33x)} = 2(cos^2 (x) -1)##
⇒ ##sin^2 (33x) = 2cos^2(x)cos^2(33x) - 2cos^2 (33x)##
⇒ ##1 - cos^2(33x)= 2cos^2(x)cos^2(33x) - 2cos^2 (33x)##
⇒ ##2cos^2(x)cos^2(33x) - cos^2 (33x) - 1 = 0##

And I end up making it more complex. It'b be great if I could convert the ##33x## to something smaller.

Please help.

Since ##\cos(2x)-1 = \cos^2(x) - \sin^2(x)-1 = 1-2 \sin^2(x)-1 = -2 \sin^2(x)##, your equation becomes ##\tan^2(33x) + 2 \sin^2(x) = 0##. When can a sum of squares equal zero?
 
  • #4
In both the ways I get the solutions as: ## \pi, 2\pi, 3\pi ... ##

So the required sum is ## = \pi + 2\pi + 3\pi +... + 99\pi = 4950\pi ##

Thanks everyone.
 

Related to Finding sum of roots of trigonometric equation

1. What is the purpose of finding the sum of roots of a trigonometric equation?

The sum of roots of a trigonometric equation is used to determine the values of the roots, which can help in solving the equation and finding its solutions.

2. How do you find the sum of roots of a trigonometric equation?

To find the sum of roots, you can use the formula: sum of roots = -b/a, where b is the coefficient of sinx or cosx term and a is the coefficient of x^2 term. Alternatively, you can also use the trigonometric identities to simplify the equation and find the sum of roots.

3. Can the sum of roots of a trigonometric equation be negative?

Yes, the sum of roots can be negative if the equation has two complex roots, which would result in a negative value when using the formula -b/a. However, if the equation has two real roots, the sum of roots will always be positive.

4. How is finding the sum of roots of a trigonometric equation useful in real life?

Trigonometric equations are used in various fields such as physics, engineering, and astronomy. Finding the sum of roots can help in solving these equations and finding the solutions, which can then be applied in real-life scenarios such as predicting the motion of objects or designing structures.

5. Are there any special cases when finding the sum of roots of a trigonometric equation?

Yes, there are two special cases: when the equation has no roots, the sum of roots will be zero. And when the equation has infinite roots, the sum of roots will be undefined.

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