Solve each right triangle. In Exercise 46, give angles to the nea...

Question

Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46.

Accepted Answer

Hey, everyone in this problem, we're asked to find the missing sidelines in the missing angle of the right triangle. OK. So we're given triangle P. QR we're told that side Q which is the hypotenuse of our triangle is 689. OK. Angle Q is a right angle. And we're also told that angle R is 47 degrees and minutes. So we're given four answer choices. Each of them containing different values for angle P side R and side. So we're gonna start working on this problem and we're gonna come back to those answer choices as we work through them. So let's start with the angle. OK. That's gonna be the easiest to do. So we're gonna start with finding angle P. OK. And recall that when we have a triangle, the sum of the angles is gonna be 180 degrees. OK. So angle P plus angle, Q plus angle R must equal degrees. We're looking for angle P as we have P, we know Q is a right angle. OK. Given in our diagram and so Q is going to be 90 degrees and we're told that R is 47 degrees and minutes. Can I recall that this single apostrophe that indicates minutes when we're talking about angles? So that's 1/60 of an angle or of a degree. All right. So 47 degrees and 35 minutes for hour and all of this is equal to 180 degrees. All right. So to find P, we just need to subtract the other two angles from this left-hand side, move them to the right hand side. We get that P is going to be equal to, well, we have 180 degrees. We subtract 90 degrees that brings us down to 90 degrees. Then we subtract 47 degrees in 35 minutes. And that is gonna give us an angle P of 42 degrees in 25 minutes. They recall that the maximum again, minute is 1/60 of a degree. And so when we're doing that subtraction, we're subtracting 35 from 60 to get this, this 25 minutes. That's left over. All right. So we found angle P 42 degrees in 25 minutes. You can look at our answer choices. Now, option A has that P is 42 degrees in 45 minutes. So that's not correct. We can eliminate that. Option B is that it is just 42 degrees. We can eliminate that as well. Option C and D both have the correct answer for angle P 42 degrees 25 minutes. So either of those could be correct. And now we're gonna move to the side lengths. Now, let's start with side link bar. What can we use here? Well, we can use our trigonometric ratios, right? We have one side length, we know these angles. So let's use our trigonometric ratio. And for R side R, we want to relate that to side Q and we're gonna use angle R. So we're gonna use sign, OK. Sign of angle R is going to be equal to the opposite side which is side R divided by the hippo, which is key. So what we have is that sign of 47 degrees in 35 minutes is equal to R divided by 689. We're gonna multiply both sides by 689. And we get that R is equal to about 508.66. And when you're typing the angles into your calculator, when they have minutes again, this is one minute is 1/60 of a degree. So here we have 47 degrees and 35/60. OK? So you can use that when you're typing that into your calculator. All right. So now we have side R 508. approximately. And we're gonna do the same thing to find side P. OK? And for side P, what we have, we're gonna do the exact same thing with coastline. OK. Because now we're talking about the adjacent angle if we use the exact same angle R or sorry, the adjacent side, if we're using the same angle, all right, cosine of R is equal to P divided by Q, the adjacent side divided by the hypo, we have cosine that same angle 47 degrees and minutes is going to be equal to side. P divided by the hypo New multiplying by both sides by 689. Gives us that side P is gonna be about 464.74. All right. So now that we have our side links, we can get back to these answer choices. We found that side R was approximately 508.7 and side P was approximately 464.7. And this is gonna correspond with answer choice D and answer choice C had those in the opposite order. So answer tray is a correct answer here. Thanks everyone for watching. I hope this video helped you in the next one.

Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46.

Key Concepts

Right Triangle Properties

Trigonometric Ratios

Angle Measurement

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