Solve each triangle. See Examples 2 and 3.

A ...

Question

Solve each triangle. See Examples 2 and 3.

A = 112.8°, b = 6.28 m, c = 12.2 m

Accepted Answer

Hello, today we are going to be finding the missing measurements of the given triangle. We are told that we have a triangle with the angle P equal to 128.3 degrees. We have the S length Q equal to 7.54 centimeters and the S length R equal to 9.30 centimeters. Let's go ahead and label this information on a generic triangle. So we have the side length Q which is 7.54 centimeters with respective angle Q, we have the side length R equal to 9.30 centimeters with the respective angle R. And we know that we have the side length P with the respective angle P which is 128.3 centimeters. How can we use this information to find the remaining angles and side length? Well, notice that we know the two sides or at least the measurements of two sides of the triangle. And we know the measurement of the angle between these two sides. We can then use the law of cosines to solve for the side length P. So let's go ahead and start by solving for the side length P. The law of cosines allows us to state that P squared is equal to Q squared plus R squared minus two, multiplied by Q multiplied by R multiplied by cosine of angle P. What we are going to do now is we're going to take the square root of both sides of this equation to solve for the side length P, we are going to have P is equal to the square root of Q squared plus R squared minus two. QR multiplied by cosine of P. Now, what we are going to do is we're going to plug in our known information. We know that the side length Q is equal to 7.54 centimeters. So we have 7.54 squared plus side length R squared, which is 9.30 squared minus two, multiplied by 7.54 multiplied by 9.30 multiplied by the cosine of angle P which is 128.3 degrees. And keep in mind that all of this is under a square root. At this step, I would recommend using a calculator to help you simplify this quantity. And after using a calculator, you'll have an approximate measurement of 15.2 m for the side length P. So this is going to be the first piece of information that we are missing from the triangle. Let's go ahead and label that. So we now know that S length P is 15.2 m. Now that we know the side length P, how can we use this information to help us solve for the angles R and Q? But what we can do is we can use the law of signs, the law of sine states that we have sine of angle P divided by si length P equal to sine of angle Q divided by S length Q equal to sine of angle R divided by the side length are what we can go ahead and do is we could take any two of the three ratios, set them equal to each other and use that to sol for the missing angles. Let's go ahead and solve for the angle R. So since we want to solve for the angle R and we know the S length R, we're going to take the ratio with respect to R. We also now know the angle P and the S length P. So we're going to use the ratio with respect to P and set it equal to the ratio with respect to R and use that to solve for the angle R. So let's go ahead and do that. So since we are solving for R using the law of sines, we are going to state that sine of angle P divided by S length P is equal to sine of angle R divided by SI length R. Now, what we want to do is we want to solve for the angle R, we can start by cross multiplying both of the ratios together that will give us S length P multiplied by sine of angle R is equal to the sil length R multiplied by sine of angle P. Since we want to solve for the angle R, we want to get it by itself on the left hand side of the equation. In order to do this, we will divide both sides of the equation by the side length P that will leave us with sine of angle R is equal to the S length R multiplied by sine of angle P divided by the S length P. Now, what we want to do is we want to remove the angle R from the sine function. In order to do this, we are going to take the sine inverse of both sides of the equation that will leave us with the angle R is equal to the sine inverse of S length R multiplied by sine of angle P divided by S length P. Now, what we are going to do is we are going to plug in our known values. So plugging in the values will give a sine inverse of the side length R which is given to us as 9.30 multiplied by sine of angle P which is given to us as 128.3 degrees divided by the side length P which we solved earlier. To be 15.2. At this step, I would recommend using a calculator to help you evaluate this quantity. And when doing so, you should end up with the approximate angle of 28.7 degrees. This is going to be the second piece of missing information that we need for the triangle. And we can now label this on our triangle. So we now know that angle R is 28.7 degrees. Now how can we use this information to solve for the angle Q? Well, we can solve for Q by using the angle sum property for triangles. The angle, some property states that the sum of all three angles within a triangle must equal 180 degrees. So let's use this property to solve for the angle Q, OK. So the angle P plus the angle R plus the angle Q must equal to 180 degrees. We know that angle P was given to us as 128.3 degrees. We had just solved for angle R which came out to be 28.7 degrees. And with the sum of angle Q she equal 180 degrees. Now, what we are going to do is we are going to suffer Q in this equation, 128.3 plus 28.7 will give us a sum of 157 degrees. So 157 plus Q must equal to 180. If we subtract 157 to both sides of the equation, we will have angle Q is equal to 23 degrees. And this is going to be the final piece of missing information that we need for the triangle. And with that being said, the answer to this problem is going to be D. 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 triangle. See Examples 2 and 3.

A = 112.8°, b = 6.28 m, c = 12.2 m

Key Concepts

Law of Sines

Triangle Sum Theorem

Ambiguous Case of the Law of Sines

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