Solve each problem. See Examples 3 and 4.

Distance through a Tun...

Question

Solve each problem. See Examples 3 and 4.

Distance through a Tunnel A tunne​​l is to be built from point A to point B. Both A and B are visible from C. If AC is 1.4923 mi and BC is 1.0837 mi, and if C is 90°, find the measures of angles A and B.

Accepted Answer

Hello, today we're going to be solving the following problem. So we are told that on a curved road, we are observing two points from point R. We are observing point B and point Q. The difference from R, I mean the distance from R to P is 2.567 kilometers and the distance from R to Q is 2.359 kilometers. We want to use this information to solve for angles P and angle Q. Now we can take the given information and imagine the information using the right triangle. We're going to quickly redraw the right triangle so that the right angle is on the left hand side of the triangle. So we're going to label the respective angles. We have point P point Q and point R which is the right angle. We know that the distance from P to R is 2.567 kilometers. And we know that the distance from Q to R is 2.359 kilometers. So how can we use this information to solve for both angles P and angle Q? Well, let's go ahead and solve for angle P. We're going to represent angle P using an unknown angle. Theta. The one thing we do know is that we have the length of the opposite side from theta. And we have the length for the adjacent side of theta. Using this information, we can use the trigonometric function tangent in order to solve for the value of data. So recall that tangent of theta is equal to the opposite side divided by by the adjacent side of a right triangle. Since we know the values of the opposite and adjacent side, we can rewrite this equation as tangent of data is equal to 2.359 divided by 2.567. Now, in order to solve for the, we're going to have to remove the from the tangent function. In order to eliminate the tangent function. On the left hand side of the equation, we're going to be taking the tangent inverse of both sides of the equation. This will allow us to rewrite the equation as theta is equal to tangent inverse of 2.359 divided by 2.567. Now, at this point, I would recommend plugging this value into a calculator when doing so, we'll end up with a continuous decimal. Theta will approximately equal to 42. 2126. And continuing, we're going to round this decimal to two decimal places. We'll take a look at the digit to the right of eight, which is given to us as two. Since this value is less than five, we're going to allow eight to remain the same. So that means theta will equal to 42. degrees. So now that we know the angle P, how can we use the angle P to sol for angle Q? Well, since we know that angle R is a right angle, angle R is equal to 90 degrees And we just solve for angle P, angle P is 42.58 degrees. Since we have the two angles out of the three angles from the right triangle, we can use the angle sum property to solve for angle Q. So angle Q states are using the angles, some property, we can say that angle Q plus angle P plus angle R will sum to 180 degrees. So let's go ahead and plug in the known values. We don't know what angle Q is. So we'll allow that to remain the same we saw for angle P and stated that angle P is 42.58 degrees. And we know that angle R is 90 degrees. Let's go ahead and solve for angle Q using this equation 42.58 plus 90 will equal to 132.58 degrees. So we have angle Q plus 132.58 is equal to degrees. Finally, in order to sol for angle Q, we're going to subtract 132.58 to both sides of the equation. This will allow us to say that angle Q is equal to 47.42 degrees. So just to reiterate, we use the angle sum property to sol for angle Q which is 47. degrees. And we use the tangent function to sulfur angle P which came out to 42.58 degrees. This is going to be the val values for angle P and angle Q. 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.

Solve each problem. See Examples 3 and 4. Distance through a Tunnel A tunnel is to be built from point A to point B. Both A and B are visible from C. If AC is 1.4923 mi and BC is 1.0837 mi, and if C is 90°, find the measures of angles A and B.

Key Concepts

Right Triangle Trigonometry

The Pythagorean Theorem

Angle of Elevation and Depression

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