Solve Trig Question: tan(2θ) = cot(φ)

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In summary, the conversation discusses a trigonometry question involving the simplification of an equation to solve for theta. Two methods are mentioned, one involving the use of the tangent and cotangent of an angle and its complement, and the other using a formula for the tangent of double an angle.
  • #1
jesuslovesu
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[SOLVED] Trig Question

Homework Statement


Let's say I have [tex]tan(2\theta) = cot(\phi)[/tex] I'm not quite sure how to simplify it (I want to solve for theta) I could say [tex]tan(2\theta) = 1 / tan(\phi)[/tex]
[tex]2\theta = atan( 1 / tan(\phi) ) [/tex] but then I'm not sure what to do after that.
Do I take the atan of 1 and [tex]tan(\phi)[/tex]?

I know another method to find it is to say [tex]cot(\phi) = \pi/2 - tan(\phi)[/tex] but I don't understand that either because [tex]2\theta = atan( \pi/2 - tan\phi ) != \pi/2 -\phi [/tex] (not exactly at least)
The answer as far as I know is supposed to be [tex]2\theta = \pi/2 -\phi [/tex]
 
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  • #2
Remember that that the tangent of angle is the cotangent of its complement and go from that definition.

OR you could say that


[tex]tan(2\theta) = cot(\phi) \Rightarrow \frac{2tan\theta}{1-tan^2\theta}=\frac{1}{tan\phi}[/tex]

and solve from there.

Two ways really, one is easier than the other though.
 
  • #3
+ n\pi where n is any integer.
To solve for theta in this equation, we can use the identities tan(x) = cot(\pi/2 - x) and cot(x) = tan(\pi/2 - x). Therefore, we can rewrite the equation as tan(2\theta) = tan(\pi/2 - \phi). Since both sides have the same trigonometric function, we can set the angles equal to each other, giving us 2\theta = \pi/2 - \phi + n\pi, where n is any integer. This is because the tangent function has a period of \pi, so we can add or subtract any multiple of \pi to the angle without changing the value of the tangent function. Therefore, the solution for theta is 2\theta = \pi/2 - \phi + n\pi, where n is any integer.
 

1. What are the possible values of θ and φ?

The values of θ and φ can range from 0 to 360 degrees or 0 to 2π radians, depending on the units being used.

2. How do I solve for θ and φ in this equation?

To solve for θ and φ, use the inverse trigonometric functions of tangent and cotangent. The inverse tangent function, arctan, can be used to find the value of θ, while the inverse cotangent function, arccot, can be used to find the value of φ.

3. Can this equation have multiple solutions?

Yes, this equation can have multiple solutions. Since tangent and cotangent are periodic functions, the values of θ and φ can repeat after every 360 degrees or 2π radians. Therefore, there can be an infinite number of solutions to this equation.

4. Is there a specific method for solving this type of trigonometric equation?

Yes, there is a specific method for solving this type of equation. It involves using the trigonometric identities of tangent and cotangent to simplify the equation and then using the inverse trigonometric functions to find the solutions for θ and φ.

5. Can this equation be solved without using trigonometric identities?

No, this equation cannot be solved without using trigonometric identities. The identities of tangent and cotangent are essential in simplifying the equation and finding the solutions for θ and φ.

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