Video Transcript
For a triangle 𝐴𝐵𝐶, 𝑎 is equal to three centimeters, 𝑏 is equal to nine centimeters, and the measure of angle 𝐴 is 10 degrees. Find all possible measures of angle 𝐵 to the nearest degree.
As we are given the lengths of two sides of our triangle, together with the measure of one of the angles, we can use the law of sines to calculate the measure of a second angle. This states that sin 𝐴 over 𝑎 is equal to sin 𝐵 over 𝑏, which is equal to sin 𝐶 over 𝑐, where uppercase 𝐴, 𝐵, and 𝐶 are the measures of the three angles of the triangle and lowercase 𝑎, 𝑏, and 𝑐 are the side lengths opposite them.
Before substituting our values into the law of sines, it is worth clarifying how many triangles exist from the given measurements. If angle 𝐴 is acute, as in this case, and the height of the triangle ℎ is less than side length 𝑎, which is less than side length 𝑏, then two triangles exist. We can calculate the height ℎ using a right triangle and the sine ratio. The sin of 10 degrees is equal to ℎ over nine. Multiplying through by nine, we have ℎ is equal to nine multiplied by sin of 10 degrees. Typing this into our calculator gives us ℎ is equal to 1.5628 and so on.
This means that the height of our triangle is 1.56 centimeters to two decimal places. And this means that ℎ is less than 𝑎, which is less than 𝑏. This confirms that there are two possible triangles 𝐴𝐵𝐶. And as a result, there are two possible measures of angle 𝐵. Substituting our measurements into the law of sines, we have sin of 10 degrees over three is equal to sin 𝐵 over nine. Multiplying through by nine, sin 𝐵 is equal to nine multiplied by sin of 10 degrees all divided by three. And entering the left-hand side into our calculator, we have sin 𝐵 is equal 0.5209 and so on.
Next, we can take the inverse sine of both sides of this equation. This gives us 𝐵 is equal to 31.39 and so on. We are asked to give our answer to the nearest degree. Therefore, one possible measure of angle 𝐵 is 31 degrees. Since the sin of 180 degrees minus 𝜃 is equal to sin 𝜃, there is another value of 𝐵 for which sin 𝐵 is equal 0.5209 and so on. Subtracting 31 degrees from 180 degrees gives us 149 degrees. This means that the second possible measure of angle 𝐵 is 149 degrees.
There are two possible triangles 𝐴𝐵𝐶 that can be formed from the measurements given in the question. And they occur when the measure of angle 𝐵 is equal to 31 degrees and 149 degrees.
