### Video Transcript

𝐴𝐵𝐶 is a triangle where the measure of angle 𝐴 is 40 degrees, side length 𝑎 is equal to five centimeters, and side length 𝑏 is four centimeters. If the triangle exists, find all possible values for the other lengths and angles in 𝐴𝐵𝐶, giving lengths to two decimal places and angles to the nearest second.

We recall that, in any triangle 𝐴𝐵𝐶, the sine rule or law of sines states that lowercase 𝑎 over the sin of angle 𝐴 is equal to lowercase 𝑏 over the sin of angle 𝐵, which is equal to lowercase 𝑐 over the sin of angle 𝐶. The lowercase letters are the lengths of the sides opposite the corresponding angles. In this question, we are told the measure of angle 𝐴 is 40 degrees, the length of side 𝑎 is five centimeters, and the length of side 𝑏 is four centimeters. We will begin by calculating the value or values of angle 𝐵. Substituting in our values gives us four over sin 𝐵 is equal to five over sin of 40.

We can flip or find the reciprocal of both of these fractions such that sin 𝐵 over four is equal to sin of 40 over five. Multiplying both sides of this equation by four gives us the sin of angle 𝐵 is equal to sin of 40 degrees divided by five multiplied by four. We can then take the inverse sin of both sides of this equation to calculate the measure of angle 𝐵. Typing this into the calculator gives us 30.946 and so on. This answer is in degrees, and we need to give our answer to the nearest second. Using the degrees, minutes, and second button on a scientific calculator gives us an answer of 30 degrees, 56 minutes, and 46 seconds.

Using our knowledge of the CAST diagram, we know that there is a second possible value of this angle between zero and 180 degrees. We can calculate this second value by subtracting 30 degrees, 56 minutes, and 46 seconds away from 180 degrees. This gives us an answer of 149 degrees, three minutes, and 14 seconds. As this value is less than 180 degrees, it seems possible it could be an angle inside our triangle. However, we were told that the measure of angle 𝐴 was 40 degrees. And as angles in a triangle sum to 180 degrees and 40 plus 149 is greater than this, this angle cannot be correct. The only possible measure of angle 𝐵 is 30 degrees, 56 minutes, and 46 seconds.

We will now use the fact that angles in a triangle sum to 180 degrees to calculate the measure of angle 𝐶. The measure of angle 𝐴 plus the measure of angle 𝐵 plus the measure of angle 𝐶 must be equal to 180 degrees. We can substitute in our values for angle 𝐴 and angle 𝐵. After adding these two values, we need to subtract 70 degrees, 56 minutes, and 46 seconds from 180 degrees. This gives us an answer of 109 degrees, three minutes, and 14 seconds.

We now need to calculate the value of side length 𝑐. Using the sine rule, once again, we have 𝑐 over the sin of 109 degrees, three minutes, and 14 seconds is equal to five over the sin of 40 degrees. We can then multiply both sides by the sin of angle 𝐶, giving us the length 𝑐 is equal to five over the sin of 40 degrees multiplied by the sin of 109 degrees, three minutes, and 14 seconds. Typing the right-hand side into our calculator gives us 𝑐 is equal to 7.3524 and so on. Rounding this to two decimal places gives us 𝑐 is equal to 7.35 centimeters.

The missing angles in triangle 𝐴𝐵𝐶 are 30 degrees, 56 minutes, and 46 seconds and 109 degrees, three minutes, and 14 seconds. And the missing side length is 7.35 centimeters.