Two sides of a triangle each measure 4 mm. The angle between these two sides has a measure of 50?. What is the length of the third side of the triangle?

Next easiest may be to divide the triangle into two right triangles, each with a hypotenuse of 4 mm and an acute angle of 50°/2 = 25°. Then half the measure of the unknown side is ...

(4 mm)sin(25°) ≈ 1.69 mm

So the measure of the entire unknown side is ...

2×(4 mm)sin(50°/2) = 2×1.69 mm = 3.38 mm

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Yet another way to find the measure is to use the law of sines. The angle at each end of the unknown side is a base angle of the isosceles triangle with a 50° apex angle. Those base angles are (180°-50°)/2 = 65°. Then the law of sines tells you ...

a/sin(A) = c/sin(C)

(4 mm)/sin(65°) = (unknown side)/sin(50°)

so, the unknown side length is ...

unknown side = (sin(50°)/sin(65°))×(4 mm) ≈ 3.38 mm

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And, the law of cosines can be used.

c² = a² + b² - 2ab·cos(C)

c² = 4² + 4² - 2·4·4·cos(50°) = 32 -32·cos(50°) ≈ 11.4308

The third side is then ...

c = √11.4308 ≈ 3.38 . . . mm