Solve each triangle. See Examples 2 and 3.

B ...

Question

Solve each triangle. See Examples 2 and 3.

B = 74.8°, a = 8.92 in., c = 6.43 in.

Accepted Answer

Hello, today we are going to be solving for the missing measurements of the given triangle. We are told that we have a triangle with the angle Q equal to 52.3 degrees. We have the side length P equal to 7.56 m and the side length are equal to 6.24 m. Let's go ahead and draw this information on a generic triangle. We're going to label this triangle with the angles Q P and R and these angles will have the respective side lengths of side length R, side length P and side length Q. Now, with the given information, we can label side length P to be 7.56 we can label the site R to be 6.24 and we can label the angle Q as 52.3 degrees. So how can we go about solving for the remaining angles and side length? Well, let's go ahead and start by solving for the side length. Little Q see in the given triangle, we know the side lengths of two sides and the angle in between these side links. Using this information, we can use the law of cosines to solve for the side length Q. The law of cosines allows us to state that the side length Q squared is equal to the sum of the sides squared which is P squared plus R squared minus two, multiplied by P multiplied by R multiplied by the cosine of the angle Q. We will use this equation to solve for the side length Q. So what we will do is we will solve for Q by taking the square root of both sides of this equation. This will allow us to write Q is equal to the square root of P squared plus R squared minus two pr multiplied by cosine of angle Q. Since we know the information, we are going to plug in the side links pr and the angle Q that is going to give us the square root of 7.56 squared plus the side length R which is 6.24 squared minus two, multiplied by 7.56 multiplied by 6.24 multiplied by the cosine of angle Q which was given to us as 52.3 degrees. Now, at this step, I would recommend plugging this into a calculator to help you solve or simplify this quantity. Once you are done using a calculator, you should end up with an approximate length of 6.20 m for the side length. QE this is going to be the first piece of missing information that we need for the triangle. We can also label this on our generic triangle. So we now know that the side length Q is approximately 6.20 m. Now, what we need to do is we need to solve for the angles R and P. Let's go ahead and start by solving for the angle P. No, we know that we have the angle Q and the S length Q we have the S length P but we are missing the angle for P. In order to solve for the angle, we can use the law of signs. Now, the law of sine states that we have three equivalent ratios. We have sine of angle P divided by P is equal to sine of angle Q divided by si length Q is equal to sine of angle R divided by the S length R. What we are going to do is we are going to pick any two of the three ratios, set them equally to each other. And use that to solve for the angle P. Now we have all the information we need to use the ratio with relate with respect to Q. Since we are solving for the angle P, we will set this ratio equal to the ratio with respect to P. So let's go ahead and solve for the angle P. Now using the law of signs, we can state that sine of angle P divided by the S length P is equal to sine of angle Q divided by the S length Q. Let's go ahead and solve for the angle P. The first thing we can do is we can cross multiply both of the ratios together that will leave us with S length Q multiplied by sine of angle P is equal to S length P multiplied by sine of angle Q. Next, what we can do since we are solving for the angle P, we want to isolate the variable angle P. So we are going to divide both sides by the side length Q doing so will leave us with sine of angle P is equal to the side length P multiplied by sine of angle Q divided by the side length Q. Now, what we want to do is you want to remove the angle P from the sine function. In order to do this, we are going to take the sine inverse of both sides of this equation. Doing so will leave us with the angle P is equal to the sine inverse of side length P multiplied by sine of angle Q divided by the side length Q. Now, what we are going to do is we are going to plug in the values for angle Q, side length Q and the side length P in order to solve for the angle P. So this is going to give a sign inverse of the side length P which is given to us as 7.56 multiplied by sine of angle Q, which was given to us as 52.3 degrees divided by the silent P which we solved earlier to be 6.20 m. At this point, I recommend using a calculator to help you solve this quantity. After plugging this quantity into a calculator, you'll get an approximate angle of 74.9 degrees for the angle P. This is going to be the second piece of information that is missing from the triangle. Now, let's go ahead and label that. So we now know that angle P is 74 0.9 degrees. Now what we are missing is angle R. Well, we can solve for angle R by using the angle. Some property for triangles recall that this angle sum property states that the sum of all three angles within a triangle must equal 180 degrees. So let's go ahead and sulfur R using the property. So that states that angle P plus angle Q plus angle R is equal to 180 degrees. We now know the measurements for angle P and angle Q, we know that angle P is 74.9 degrees plus angle Q which was given to us as 52.3 degrees plus angle R is equal to 180 degrees. Let's go ahead and simplify this equation to sulfur angle R 74.9 plus 52.3 will give us 127.2. So we have 127.2 plus angle R is equal to 180 degrees. We will subtract 127.2 to both sides of this equation. And this will leave us with the angle R is equal to 52.8 degrees. And this is going to be the final piece of missing information that we need for our triangle. 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.

Solve each triangle. See Examples 2 and 3.

B = 74.8°, a = 8.92 in., c = 6.43 in.

Key Concepts

Law of Sines

Triangle Properties

Angle of Elevation and Depression

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