Calculus/Question

Advertisement


Question
Okay, will keep my question as straightforward as possible:
Solve the following equation algebraically over the interval 0 is equal to or less than x < 2pi and round the answer to the nearest hundredth of a radian.

The equation is: csc^(2)x=5

Thank you for your help.

Answer
Hi John,
The easy way to do this would be by using a Maclaurin series expansion.
We have
csc²(x) = 5
csc(x) = √5
and
sin(x) = 1/√5 ≈ ±0.4472
Now,
sin(x) = x - (x^3)/6 + (x^5)/120 + .......
The function, x - (x^3)/6 + (x^5)/120, approximates sin(x) very well in the interval 0 < x < π/2 but x - (x^3)/6 also does well and is easier to solve so we'll use it.
Solving for the positive value,
x - (x^3)/6 = 0.4472
which gives x ≈ 0.464 or just 0.46 and lies in the required interval.
We do not need to solve for the negative value since the function is odd and we know that x = -0.46 would also be a solution but is outside our interval. We can always use the solution that lies in the interval 0 < x < π/2 to find the other solutions.
In general, for
sin(A) = B
where 0 < A < π/2 and B > 0 (as it has to be), then
sin(π - A) = B
sin(π + A) = -B
sin(2π - A) = -B
And so the complete solution to sin(x) = ±0.4472 would be
x = 0.464, π - 0.464, π + 0.464, 2π - 0.464
and to the nearest hundredth
x = 0.46, 2.68, 3.61, 5.82
and are consequently the approximate solutions to csc²(x) = 5

We could also have used the Maclaurin series for csc(x) directly, since
csc(x) = (1/x) + x/6 - 7x³/360 + ...
or just (1/x) + x/6 in the interval 0 < x < π/2 as before.

I do not know what you're studying exactly but I see here that it's in calculus so i supposed that how to solve a cubic equation isn't the issue here and have therefore skipped the details of that. You can always get back to me if anything is unclear.

Regards

Calculus

All Answers


Answers by Expert:


Ask Experts

Volunteer


Ahmed Salami

Expertise

I can provide good answers to questions dealing in almost all of mathematics especially from A`Level downwards. I believe i would be very helpful in calculus and can as well help a good deal in Physics with most emphasis directed towards mechanics.

Experience

Aspiring theoretical physicist. I have been doing maths and physics all my life.

Education/Credentials
I teach mathematics and engineering physics.

©2016 About.com. All rights reserved.