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You need the Law of Cosines here.  Let's call our given angle A, so the side across from it, the one we are looking to solve for, would be a.  The Law is as follows then: [tex]a^2=27^2+31^2-[2(27)(31)cos(75)][/tex].  We will simplify everything here at once to get [tex]a^2=729+961-433.26308[/tex].  [tex]a^2=1256.73692[/tex] so a = 35.5

[tex]35.5[/tex] mm is the approximate length of the third side of the triangle below.

What is triangle?

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane.

What is cosine law?

In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles

According to question, we have 31mm and 27mm as two sides of triangle and 75° is the angle between them.

We have to find the length of the third side of the triangle.

Let [tex]a[/tex]  be the length of third side.

Applying the cosine law, we will get

[tex]a^{2} =27^{2} +31^{2} -2(27)(31)cos75[/tex]°

⇒[tex]a^{2} =1256.53[/tex]

⇒[tex]a=35.5[/tex]

Hence,[tex]35.5[/tex] mm is the approximate length of the third side of the triangle below.

Learn more about triangle here:

https://brainly.com/question/20063785?referrer=searchResults

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