In Exercises 85–96, use a calculator to solve each equation, c...

Question

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0

Accepted Answer

Hello, today we're going to be evaluating the following trigonometric equation with the help of a calculator. So what we are given is six tangents squared of X minus seven tangent of X minus two is equal to zero. Now, before we use the calculator, we're first going to need to simplify the given equation. Notice that the left hand side of the equation roughly resembles a polynomial. We can simplify the left hand side by using a substitution. Let's use Y is equal to tangent of X. If we apply this substitution, we can rewrite the left hand side of the equation as six Y squared minus seven, Y minus two is equal to zero. Currently, this is not a factor trinomial, but we can solve for Y by using the quadratic equation. If we use the quadratic equation, we get Y is equal to seven plus or minus the square root of negative seven squared minus four multiplied by six, multiplied by negative two, all over two multiplied by six. Now, in the numerator negative seven squared will give us the value of and negative four multiplied by six, multiplied by negative two will give us the value of positive 48. So we have Y is equal to seven plus or minus the square root of 49 plus 48 all over six times two, which will give us 12. Now, 49 plus 48 will give us the value of 97. So Y is going to equal to seven plus or minus the square root of 97 all over 12. Now, this means that Y is equal to two separate values, Y will equal to seven plus the square root of 97/12. And Y will equal to seven minus the square root of 97/12. And if we were to ress substitute Y with tangent of X, we can rewrite these values as tangent of X is equal to seven plus the square root of 97/12. And tangent of X is equal to seven minus the square root of 97/12. Now, keep in mind that we only want the values of X that lie within the range of 0 to 2 pi and that means the values of X can only be positive. So we're going to be ignoring the negative value of tangent of X. So now the question is, what value of X can we plug in a tangent that will give us the value of seven plus the square root of 97/12. This is where we'll need to plug in the equation into a calculator. If you plug in the equation to a calculator, you will end up with four different values of X. But keep in mind that we only want these values up to four decimal places. So using a calculator, the four estimated values of X that we get for tangent of X is going to be X is equal to 0.9519, X is equal to 2.9085. X is equal to 4. and X is equal to 6.501. These are going to be the four estimated values of X within the region of 0 to 2 pi and with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0

Key Concepts

Solving Quadratic Equations in Trigonometric Functions

Using the Tangent Function and Its Properties

Finding Solutions on a Specified Interval with a Calculator

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