Solve each triangle. See Examples 2 and 3.

a ...

Question

Solve each triangle. See Examples 2 and 3.

a = 3.0 ft, b = 5.0 ft, c = 6.0 ft

Accepted Answer

Hello, today we are going to be finding the missing measurements of the triangle. We are told that we have a triangle where the side length P is equal to seven inches, the side length Q is equal to eight inches and the side length R is equal to nine inches. Let's go ahead and label this information on a generic triangle. Now, we know that we have the side links, PQ and R. These side links will also have the respective angles of angle P angle Q and angle R. We were told that the side length P is equal to seven inches. The side length Q is equal to eight inches and the side length R is equal to nine inches. How can we use this information to solve for the missing angles? Well, since we know the measurements of all three side links, we can use this information and the law of cosines to solve for one of the missing angles. Let's go ahead and start by solving for the angle R. Now using the law of cosines, the law of cosines allows us to state that the side length R squared is equal to the side length P squared plus the side length Q squared minus two multiplied by side links, P and Q multiplied by cosine of the angle R. Since we want to solve for the angle R, let's go ahead and get angle R by itself. On the right hand side, we can start by subtracting P squared and Q squared to both sides of the equation that will allow us to write R squared minus P squared minus Q squared is equal to negative two multiplied by P multiplied by Q multiplied by cosine of angle R. What we can do now is we can eliminate the negative sign by dividing both sides by negative one doing so will allow us to write P squared plus Q squared minus R squared is equal to two multiplied by P multiplied by Q multiplied by cosine of angle R. Now, what we want to do is we want to eliminate the product of two PQ. On the right hand side of the equation, we can do this by dividing both sides of the equation by two PQ. That will allow us to write that the cosine of angle R is equal to P squared plus Q squared minus R squared divided by two multiplied by P multiplied by Q. Since we want to solve for the angle R, we want to remove the angle R from the cosine function. In order to do this, we will be taking the cosine inverse of both sides of the equation that will leave us with the angle R is equal to the cosine inverse of P squared plus Q squared minus R squared divided by two multiplied by P multiplied by Q. Now that we have an equation for the angle R, let's go ahead and plug in our known information. So we have the cosine inverse of the side length P squared, which is seven squared plus the side length Q squared, which is eight squared minus the side length R squared, which is nine squared divided by two multiplied by sil length P which is seven multiplied by S length Q which is eight. At this point. I would recommend using a calculator to help you evaluate this quantity. When doing. So you will get an approximate angle of 73 degrees, which is going to be the first piece of missing information that we need for our triangle. Now let's go ahead and label this information on our triangle. So we know that angle R is approximately equal to 73 degrees. So now that we know the side length are the angle R, how can we use this information to help us solve for the angles Q and P? But what we can do is we can use the law of signs to solve for one of the other missing angles. 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 si length R. We can use this information to then solve for the angle P. What we are going to do is we are going to take the ratio with respect to P and set it equal to the ratio with respect to R since we know the S length R and the angle R, so let's go ahead and solve for the angle P. So we have sine of angle P divided by S length P is equal to sine of angle R divided by the S length R. We can start for tool for the angle P by cross multiplying both of the ratios together. This will give us the S length R multiplied by sine of angle P is equal to the S length P multiplied by sine of angle R. Since we want to solve for the angle P, we are going to divide both sides of the equation by the side length R, this will allow us to write sine of angle P is equal to the side length P multiplied by sine of angle R divided by the side length R. Since we want to solve for the angle P, we need to remove that from the sine, from the sine function to do this, we will need to take the sine inverse of both sides of the equation that will leave us with the angle P is equal to the sine inverse of the side length P multiplied by sine of angle R divided by the side length R. Now, what we are going to do is we are going to plug in our known information. So this will give us sine inverse of the si length P which is seven multiplied by sine of angle R which we solved to be 73 degrees divided by the S length R which is nine. I would recommend using a calculator to help you evaluate this quantity. And you should get an approximate angle of 48 degrees. This is going to be the second piece of missing information for our triangle. And we can label this on our generic triangle. So we now know that the angle P is 48 degrees. So now that we have the two angles, how can we use this information to solve for the angle Q? While in order to solve for Q, we can use the angle property, the angle sum property states that the sum of all three angles within a triangle must equal 180 degrees. So the property states that the sum of angle P with the sum of angle Q with the sum of angle R must equal to 180 degrees. Let's go ahead and use this information or use this equation to solve for the angle Q. So we know that the angle P is equal to 48 degrees. We are adding angle Q and we know that angle R is 73 degrees, 48 plus 73 will give us a sum of 121. So 121 plus angle Q is equal to 180 degrees. And if we subtract 121 from both sides of the equation, we will get an angle Q is equal to 59 degrees. 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 B 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 = 3.0 ft, b = 5.0 ft, c = 6.0 ft

Key Concepts

Law of Cosines

Law of Sines

Triangle Inequality Theorem

Watch next