Step 1: Recall the formula to calculate area of the triangle.

            NOTE: If you know the measures of two angles and the length of one's

            opposite side, you can use the Law of Sines to solve for the length of the

            other angle's opposite side.

                 FORMULA: \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}


               FORMULA: The area of  \triangle ABC is half of the product of

                   two side lengths and the sine of their included angle.

                       Area = \frac{1}{2}bc \sin (A)

                       Area = \frac{1}{2}ac \sin (B)

                       Area = \frac{1}{2}ab \sin (C)

              Where a, b, and c are the lengths of the sides opposite the angles A, B, and C.

Step 2:  Find the remaining angle in the triangle

              NOTE: The sum of the angles in a triangle is 180\degree

                             \angle T + \angle U + \angle V = 180\degree

                             20\degree + \angle U + 41\degree = 180\degree

                             \angle U = 180\degree - 61\degree

                             \angle U = 119\degree

Step 3: Use the Law of Sines to find the length of the side opposite(t) to an angle ( \angle T).

              Law of sine

                   \frac{v}{\sin V} = \frac{t}{\sin T}

                   \frac{42}{\sin 41\degree} = \frac{t}{\sin 20\degree}

                   \frac{42}{0.6560} = \frac{t}{0.3420}

                   64.0186 = \frac{t}{0.3420}

                   t = 21.8956


Step 4: Find the area of the triangle

            NOTE:  We know

                                 Area = \frac{1}{2}tv \sin U

                                 Area = \frac{1}{2}21.8956*42 \sin 119\degree

                                 Area = \frac{1}{2}21.8956*42 *(0.8746)

                                 Area = 402.1579…

                        The area of \triangle TUV is 402.2 square meters.