What is the positive value of D D D in the interval [ 0 , π 2 ) \left[0,\frac{\pi}{2}\right) [ 0 , 2 π ) that will make the following statement true? Express the answer in four decimal places. sec D = 3.2842 \sec D=3.2842 sec D = 3.2842

Let's see how we can solve this problem. So in this problem, we are asked what is the positive value of d in the interval zero to pi over two that will make the following statement true. And our statement says that the secant of d is equal to 3.2842. We're We're asked to express the answer in four decimal places. Okay. So to solve this problem, we're going to need to find a way to isolate d. Now something I noticed is that we're dealing with a secant, and there isn't really any kind of secant or inverse secant operations on our standard calculator. So just leaving it like this isn't gonna help us much. But I recall from the reciprocal identities that secant d is the same thing as one over cosine d, because one over cosine gives you the secant. So we can set one over cosine d equal to 3.2842. And since I know there's a cosine and an inverse cosine operation on the calculator, this is a much better position to be in. Now what I'm gonna do from here is try to isolate the cosine of d by itself. And I'm gonna multiply cosine d on the left side and on the right side. Let's look at the cosine d's to cancel on the left, leaving us with just one, and this is going to equal 3.2842 multiplied by the cosine of d. From here, I can divide 3.2842 on both sides of this equation. And this will allow me to isolate the cosine d by itself, so these numbers will cancel here, giving us that the cosine of d is equal to one divided by 3.2842. Now next, I'm gonna go ahead and take the inverse cosine on both sides of this equation, and this will allow me to isolate the d by itself. So notice that the inverse cosine and the cosine will cancel, leaving me with just d. And d is going to equal the inverse cosine of this fraction that we have right here, which is one divided by 3.2842, and then that's gonna be our fraction. And what we can do from here is plug this into a calculator to see what we get. But this is a situation where you have to be careful, because notice that our interval is from zero to pi over two. Whenever you see pi in these types of fractions, that means we're dealing with radians. So you wanna make sure your calculator is in radian mode when you calculate this. So if you've been dealing with degrees so far, go to your calculator, click on mode, and that'll take you to a menu. Make sure that you select radians when you're on that menu. So once you've done that, you can go ahead and type this value into your calculator. You can press second and then the cosine button. That'll give you the inverse cosine, and then put this fraction into your calculator. If you do this, you should get that d is approximately equal to 1.261395304, then this decimal keeps going on. But we can see here that we're asked to express our answer in four decimal places. So to do this, what I'm gonna do is say that d is equal to one point, and then we're gonna save these four decimal places, which is two, six, one. And because we have a nine right here, I'm actually gonna round this three up and say that that's four. So this is our value for d, and that is the answer to the problem. So this is the solution, and I hope you found this video helpful. Thanks for watching.

12. Trigonometric Functions

Trigonometric Functions on Right Triangles

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on Right Triangles

Struggling with Business Calculus?

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Multiple Choice

What is the positive value of $D$ in the interval $\left[0,\frac{\pi}{2}\right)$ that will make the following statement true? Express the answer in four decimal places.
$\sec D=3.2842$

0.3094 rad

1.2614 rad

0.4760 rad

1.0934 rad

Verified step by step guidance

Step 1: Recall the definition of the secant function. The secant function is defined as sec(D) = 1/cos(D). This means that if sec(D) = 3.2842, then cos(D) = 1/3.2842.

Step 2: Calculate the value of cos(D) by taking the reciprocal of 3.2842. This will give you cos(D) = 1/3.2842.

Step 3: Use the inverse cosine function (arccos) to find the angle D. Since cos(D) = 1/3.2842, D = arccos(1/3.2842).

Step 4: Ensure that the value of D is within the given interval [0, π/2). The arccos function naturally returns values in this range, so the result will be valid.

Step 5: Once you compute the arccos value, round the result to four decimal places to match the problem's requirements.

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Struggling with Business Calculus?

What is the positive value of DDD in the interval [0,π2)\left[0,\frac{\pi}{2}\right)[0,2π) that will make the following statement true? Express the answer in four decimal places.secD=3.2842\sec D=3.2842secD=3.2842

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Trigonometric Functions on Right Triangles practice set

What is the positive value of $D$ in the interval $\left[0,\frac{\pi}{2}\right)$ that will make the following statement true? Express the answer in four decimal places.
$\sec D=3.2842$