In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

Video transcript

Hello, today we're going to be using the given angle and side lengths to find the remaining angles and side lengths of the triangle. And if there's no triangle that exists, we're going to state that there is no triangle. So we have, we're going to start by drawing a generic triangle. Now, a triangle has a total of three angles. We're going to let label the angles capital A capital B and capital C across from angle C will be side length, little C across from capital A will be side length, little A and across from capital B will be side length, little B. Now we are told that little A is equal to 19. We are told that little B is equal to nine and we are told that angle C is equal to 71 degrees. So we're going to use this information to find angles, capital A capital B and side length C. Well, one thing to note is that since we are given two side links and one angle, we can start by using the law of cosines to find the side length C using the law of cosines, the law of cosine states that the side length C will be C squared is equal to the side length A squared plus the side length B squared minus two, multiplied by sine length A multiplied by sine length B. And that will be multiplied by cosine of angle capital C. So using the law of cosines, let's start by solving for the side length C. Now we have the values for A B and the angle C. So we can start by plugging in those values. And that will give us C squared is equal to squared plus nine squared minus two, multiplied by 19, multiplied by nine, multiplied by cosine of 71 degrees. We're going to evaluate 19 squared, which will give us 361. So we have C squared is equal to 361 plus nine squared, which is 81 minus two multiplied by multiplied by nine, which will give us negative 342 multiplied by cosine of 71 degrees. Then what we are going to do is we're going to add 361 with 81 that will give us C squared is equal to 442 minus 342 multiplied by cosine of degrees. In order to solve for C, we're going to have to take the square root of both sides of the equation that will leave us with C is equal to the square root of 442 minus multiplied by cosine of 71 degrees. Now, at this point, I would recommend plugging in this value into a calculator and plugging in the value into a calculator will give you an approximate value for the side length C which is 18.18394. Now we're going to estimate this value to one decimal place, the digit to the right of one is eight. So we're going to round one up by one unit. So the approximate value for C is C is equal to 18.2. This will be the side length for C. Now that we have the sign length for C, we're going to want to find angles A and angles B. So now that we have the three side links and we have at least one of the angles, we can utilize the law of signs to help us find the remaining side links or side angles. So the law of sines states that if you have sine of angle A divided by s side length A that is equal to sine of angle B divided by B which is equal to sine of angle C divided by side length C. So what we can do is we can take any combination of these ratios, set them equal to each other and use the two ratios to help us solve for angle A or angle B well, we have S length A, I'm sorry, we have angle C and side length C. So let's start by choosing sign of C over C. And now we're going to choose an angle to solve for, let's go ahead and start by solving for angle A. So we don't know the measurement of angle A. So we're gonna get, we're going to set S sign of C over C equal to sign of A divided by the side length of A. We know that the side length of A is equal to 19. We now know the value of side length C which is 18.2. And we know that the angle of C is 71. So if we plug in these values, we get sign of a divided by 19 is equal to sign of 71 divided by 18.2. In order to solve for angle A, we're going to multiply both sides of the equation by 19. That will leave us with sign of A is equal to 19 multiplied by sine of 71. And that will be divided by 18.2. Next, we need to get a out of the sine function. And in order to do that, we can multiply both sides of the equation by sine inverse. That will leave us with A is equal to the sine inverse of 19 multiplied by sine of divided by 18.2. Now, at this point, I would recommend plugging in this value into a calculator. And if you plug in the value of angle A into a calculator, you get angle A is approximately equal to 81.09695. We're going to estimate this value to one decimal place. So we're going to pay attention to the digit to the right of zero. Since nine is greater than five, we're going to round zero up one unit and the approximate angle A is equal to 81.1 degrees. So we're going to label that on our generic triangle. Now that we have the angle A, we can use angle A and angle C to find the value for angle B. Now recall that the sum of the three angles in A triangle has to add up to give us the value of 180 degrees. We know that angle C is equal to 71 degrees. And we know that angle A is equal to 81.1 degrees. If we use these angles, we can plug it into the equation to sulfur angle B. So that will give us 180 is equal to 81.1 degrees plus the angle B plus 71 degrees. 81.1 plus 71 will give us the value of 152.1. So we have 180 degrees equal to 152.1 plus the angle B, we're going to subtract 152.1 from both sides of the equation. And that will give us angle B is equal to 27. degrees. So that is going to be the measurement of the final angle. So just to summarize, we were given angle C as 71 degrees and we were given the site links A and B which is equal to 19 and nine. And utilizing the law of signs and the law of cosines, we were able to calculate angles A as 81.1 angle B as 27.9 and the side length C as 18.2. And with that being said, the answer to this problem is going to be C 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.