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 0 to pi over 2 that will make the following statement true. And our statement says that the secant of d is equal to 3.2842. We're asked to express the answer in 4 decimal places. Okay. So to solve this problem, we're going to need to find a way to isolate d. Now something I notice is that we're dealing with a secant and there isn't really any kind of secant or inverse secant operations on our standard calculators. 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 1 over cosine d because one over cosine gives you the secant. So we can set 1 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 1, and this is going to equal 3.28 42 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 1 divided by 3.2842. Now next I'm going to 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 1 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 0 to pi over 2. 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 1.26139 5304, then this decimal keeps going on. But we can see here that we're asked to express our answer in 4 decimal places. So to do this, what I'm gonna do is say that d is equal to 1 point, and then we're gonna save these 4 decimal places, which is 261. And because we have a 9 right here, I'm actually gonna round this 3 up and say that that's 4. 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.

8. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

Precalculus

8. Trigonometric Functions on Right Triangles

Trigonometric Functions on Right Triangles

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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

Understand that the problem is asking for the angle D in radians such that sec(D) = 3.2842, where D is in the interval [0, π/2).

Recall that the secant function is the reciprocal of the cosine function, so sec(D) = 1/cos(D). Therefore, we need to find D such that 1/cos(D) = 3.2842.

Rearrange the equation to find cos(D): cos(D) = 1/3.2842.

Use a calculator to find the angle D in radians by taking the inverse cosine (arccos) of the value obtained in the previous step: D = arccos(1/3.2842).

Ensure that the calculated angle D is within the specified interval [0, π/2). If it is not, adjust accordingly, but in this case, the angle should naturally fall within the interval.

6:4m

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Struggling with Precalculus?

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.sec⁡D=3.2842\sec D=3.2842secD=3.2842

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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.
$\sec D=3.2842$