Final answer:
To find the two possible values for angle c in the triangle, we use the Law of Sines. After setting up the ratio and solving for sin C, we find that the possible values for angle C are approximately 23.6° and 53.1° when rounded to the nearest tenth of a degree.
Step-by-step explanation:
The question involves finding two possible values for angle c in a triangle with angle b = 19°, side b = 12, and side c = 22. Since the sum of angles in a triangle is 180°, and we have one angle, we can use the Law of Sines to find the possible values for angle c. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
Using the Law of Sines:
Calculate the ratio for angle B:
(sin 19°) / 12 = (sin C) / 22
Solve for sin C:
sin C = (sin 19° / 12) * 22
Calculate the value of sin C and then find angle C using the inverse sine function, keeping in mind that there could be two possible angles since the sine function is positive in both the first and second quadrants.
The two possible values for angle C, calculated to the nearest tenth of a degree, are 23.6° and 53.1°.