Solve each right triangle. When two sides are given, give angl...

Question

Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.

Right triangle XYZ with hypotenuse 89.6 cm, angle 47.8°, and right angle at Z, sides labeled x and y.

Accepted Answer

Hey, everyone in this problem, we're asked to find the missing sidelines in the missing angle in the triangle that we're given. So we're given a right triangle. OK. Angle Q is the right angle. We have a hypotenuse of 78.8 centimeters. And we're told that angle R is 41.4 degrees. OK. And this is triangle P. QR we have four answer choices each giving different values for angle P and side P and R. And we're gonna come back to those as we work through this problem. So the first thing we're gonna do is find this missing angle P. That's gonna be the easiest part of this problem. And we're gonna do that by remembering that the sum of the angles in a triangle is 180 degrees. OK. So angle P was a, the last angle R is going to be equal to 180 degrees. Now, we're looking for angle P, so we have P angle Q which is degrees. And we can see that in our diagram plus angle R which is 41.4 degrees. This is all equal to 180 degrees to isolate for P, we're gonna subtract our 90 degrees and our 41.4 degrees from both sides and we get that angle P is equal to 0.6 degrees. So we're done one part of this question. We found the missing angle. If we look at our answer choices, we can already eliminate. Option D, it had the missing angle to be 58.6 degrees, which is not correct. So we'll eliminate D and we'll leave option A B and C because they have the right answer for angle P. Now let's move on to these missing side links. Let's start with the missing side. R. Now, what we can do is use our trigonometric ratios. OK. This is a right angle triangle. We can use our trigonometric ratios. We have the angle 41.4 we have the hypotenuse. So let's go ahead and use sign, you recall that sign of theta is going to be equal to the opposite side divided by the hypo. In this case, we have sine of 41.4 degrees is equal to the opposite side, which is R divided by the hypo 78.8 centimeters. And we're gonna multiply both sides by 78.8 centimeters. And that's gonna give us that R it's equal to about 0.1114 centimeters. All right. So we found side R, we can do the exact same thing for IP now for side P instead of using sign of the angle, we're gonna use cosine of the angle. Why? Because now we're talking about the adjacent side. OK. So what we have is cosine of data is equal to the adjacent side divided by the hypotenuse. And so in this case, cosine of 41.4 degrees is equal to side P that we're looking for divided by the hypotenuse of 78.8 centimeters. And this gives us a sideline pee that is approximately equal to 59. years. All right. So looking back at our answer choices, we went round to one decimal place. We found that P is approximately 59.1 centimeters and R is approximately 52.1 centimeters, which is gonna correspond with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

Solve each right triangle. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2.

Key Concepts

Pythagorean Theorem

Trigonometric Ratios (Sine, Cosine, Tangent)

Angle Measurement in Degrees and Minutes

Watch next