# Finding all values that satisfy the equation

• EL ALEM
In summary, the conversation discusses finding all values of x in the interval [0,2pi] that satisfy the equation tan^2(x)-1=(2/sqrt3)tan(x). The conversation mentions using the quadratic formula to find exact answers, and the idea of using trig substitutions for more complicated solutions. It is also mentioned that the quadratic formula works for quadratics in terms of u, where u=f(x). The conversation also discusses finding all values for x in the interval using the general solution of tan(x)=y, and how this is different from finding values for sin and cos.
EL ALEM

## Homework Statement

Find all values of x in the interval [0,2pi] that satisfy:

tan^2(x)-1=(2/sqrt3)tan(x)

## The Attempt at a Solution

Is there anyway to this without using the quadratic formula that will get me exact answers?

Not that I know of, but if there is (there most likely is) then it would involve trig substitutions and such and would involve much more work and complications than just using the quadratic formula.

Yes, of course it will! Do you trust the quadratic formula when the quadratic is in terms of x? Well then it works for quadratics in terms of u where u=f(x), in this case, u=tan(x).

i think the quadratic formula would be the easiest...
And the exact answers as you say can be found only for a particular interval [-pi/2,pi/2]...or else you get a general solution...

Yup thanks for the reply, I ended up doing using the quadratic formula and ended up simplifying nicely (i got tanx= -sqrt3/3, sqrt3)

but one question how would i find ALL the values on [o,2pi], i know how to find the ones in the first quadrant (i.e. tanx= sqrt3 => x=pi/3 ) but how do i find the rest? I forget how to do that.

you know general soln of tan(x)=y is
x = tan-1(y) + k*pi , where k can be any integer.

Ok is it the same thing for sin and cos? thanks in advance.

did you understand it?

Yup that cleared A LOT of things up. Thanks a mill.

## 1. What does it mean to "find all values that satisfy the equation"?

When we say "find all values that satisfy the equation," we are referring to finding all possible solutions or values of a given equation. This means that we are looking for all the possible values of the variable(s) that make the equation true.

## 2. How do I find all values that satisfy the equation?

Finding all values that satisfy an equation involves manipulating the equation and solving for the variable(s). This may involve using algebraic techniques such as factoring, substitution, or the quadratic formula. It is important to carefully follow the rules of algebra and double-check your work to ensure that all possible solutions are found.

## 3. Are there any specific steps to follow when finding all values that satisfy an equation?

While the specific steps may vary depending on the equation and the techniques used, there are some general guidelines to follow when finding all values that satisfy an equation. These include simplifying the equation, isolating the variable, and checking for extraneous solutions. It is also important to carefully follow the order of operations and to check your answer by plugging it back into the original equation.

## 4. What if I cannot find all the values that satisfy the equation?

In some cases, it may not be possible to find all values that satisfy an equation. This may be due to the complexity of the equation or the lack of techniques available to solve it. In these cases, it is important to clearly state the limitations of the solution and to provide the values that were able to be found.

## 5. Can I use a graphing calculator to find all values that satisfy the equation?

Yes, a graphing calculator can be a helpful tool when trying to find all values that satisfy an equation. By graphing the equation, you can visually see the points where the equation intersects the x-axis, which represent the solutions. However, it is still important to carefully check your answers and to show your work when solving equations, even with the use of a calculator.

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