Final answer:
To find the angle ∆f of a triangle, the Law of Sines can be used, relating the sides of the triangle to the opposite angles. Angle ∆f can be calculated by the formula 78/sin(∆f) = 39/sin(12°) and solving for sin(∆f) followed by finding the inverse sine.
Step-by-step explanation:
To find ∆f, we can use the Law of Sines, which relates the lengths of sides of a triangle to the sines of the opposite angles. The formula is a/sin(∆A) = b/sin(∆B) = c/sin(∆C), where a, b, and c are the lengths of the sides of the triangle, and ∆A, ∆B, and ∆C are the opposite angles. Given f = 78 cm, g = 39 cm, and ∆h = 12°, we can write:
78/sin(∆f) = 39/sin(12°)
By solving for sin(∆f), we get:
sin(∆f) = 78 * sin(12°) / 39
Calculate the sine value and then use the inverse sine function (sin⁻¹) to find the angle ∆f. The calculation will yield the angle to the nearest degree.
Note that there's a possibility of having two different angles (acute and obtuse) that satisfy the sine equation. However, since the sum of angles in any triangle is always 180°, we can subtract the known angles from 180° to find ∆f if necessary.