Solving cot x = (3)^(1/4): pi/6 & 5pi/6

  • Thread starter Aaron H.
  • Start date
In summary, There are two solutions to the equation cot x = (3)^(1/4): pi/6 and 5pi/6. To solve this equation without a calculator, you can use the trigonometric identity cot x = 1/tan x and take the inverse tangent and inverse cotangent of (3)^(1/4). The reason for two solutions is because cotangent is a periodic function. You can also use the Pythagorean identity sin^2 x + cos^2 x = 1 to solve for sin x and cos x and then use inverse trigonometric functions to find x. To check if your solutions are correct, you can use a calculator or graphing calculator.
  • #1
Aaron H.
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0

Homework Statement



Solve, finding all solutions in [0, 2pi)

Homework Equations



sqrt (cot x) = (3)^(1/4)

The Attempt at a Solution



[sqrt (cot x) = (3)^(1/4)]^4 =

sqrt [cot^2 x = 3] =

cot x = +/- sqrt (3)

x = pi/6, 5pi/6, 7pi/6, 11pi/6

the answer choices to this problem only have two angles each, so two of the angles I derived aren't necessary
 
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  • #2
Since the original problem has (cot x), the solution can't include the values corresponding to cot x = -√3.
 

1. What is the solution to cot x = (3)^(1/4)?

There are two solutions to this equation: pi/6 and 5pi/6.

2. How can I solve cot x = (3)^(1/4) without a calculator?

To solve this equation without a calculator, you can use the trigonometric identity cot x = 1/tan x. Then, you can solve for tan x by taking the inverse tangent of both sides. Finally, solve for x by taking the inverse cotangent of (3)^(1/4).

3. Why are there two solutions to this equation?

Since cotangent is a periodic function, it repeats itself every pi radians. Therefore, there are multiple angles that satisfy the equation cot x = (3)^(1/4).

4. Can I use a different method to solve this equation?

Yes, you can also use the Pythagorean identity sin^2 x + cos^2 x = 1 to solve for sin x and cos x. Then, you can use the inverse sine and cosine functions to find the values of x.

5. How can I check if my solutions are correct?

You can use a calculator to check if your solutions are correct. Simply plug in the values of pi/6 and 5pi/6 into the original equation cot x = (3)^(1/4) and see if they give you the same value on both sides of the equation. Alternatively, you can also use a graphing calculator to graph both sides of the equation and see where they intersect.

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