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

Question

In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅).

cos² x - cos x - 1 = 0

Accepted Answer

Hello, today we're going to be solving the trigonometric equation with the help of a calculator. So what we are given is sine squared of X minus sign of X minus one is equal to zero. Now, before we use a 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 this by using a substitution. Let's say Y is equal to sign of X. If we were to apply this substitution to the left-hand side of the equation, we can repre we can rewrite the equation as Y squared minus Y minus one is equal to zero. Now, currently, this is not in the form of a factor trinomial, but we can solve for Y by using the quadratic equation. If we use the quadratic equation Y is going to equal to one plus or minus the square root of negative one squared minus four multiplied by one multiplied by negative one, all over two, multiplied by one in the numerator negative one squared will give us positive one and negative four multiplied by one multiplied by negative one will give us positive four. So Y is equal to one plus or minus the square root of one plus four all over two, multiplied by one, which is two. Now, one plus four is going to give us five. So we can rewrite this statement as Y is equal to one plus or minus the square root of 5/2. And this means we have two values of Y Y is equal to one plus the square root 5/2 and Y is equal to one minus the square root of 5/2. Now recall that we stated earlier that Y is equal to sin of X. If we were to ress substitute sin of X for Y, we can rewrite these equations as sin of X is equal to one plus the square root 5/2 and sign of X is equal to one minus the square root 5/2. Now recall that we only want the values of X that lie in the region of 0 to 2. Pi what this means is that we only want positive values for X and since we only want positive values for X, we're going to be ignoring the negative option for sign of X. So what we are left with is sign of X is equal to one plus the square root 5/2. And the question here is what values of X can we plug in as sign in order to give us the value of one plus square root 5/2. This is where we're going to need the calculator. If we were to plug in this equation into a calculator, we'll end up with two values of X and recall that we want these values to be estimated to four decimal places. So using a calculator, the two values of X that we get is X is equal to 3.8078 and X is equal to 5.6169. These are going to be the estimated values of X that lie 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). cos² x - cos x - 1 = 0

Key Concepts

Trigonometric Functions

Quadratic Equations

Interval Notation

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